OK  FIT   OIF 


1 1  •    i'racy  Crawford. 


ASTRONOMT  DEW: 


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^**M**^J*4     Mn-  H> 


WORKS  OF 
PROF.  W.  WOOLSEY  JOHNSON 

PUBLISHED    BY 

JOHN  WILEY  &  SONS. 

An  Elementary  Treatise  on  the  Integral  Calculus. 

Founded   on    the    Method    of    Rates.     Small  8vo, 
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Curve  Tracing  in  Cartesian  Coordinates. 

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Differential  Equations. 

A  Treatise  on   Ordinary  and   Partial   Differential 
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$3-50. 
Theoretical  Mechanics. 

An  Elementary   Treatise.     i2mo,  xv-f-434  pages, 
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An  Elementary  Treatise  on  the  Differential  Cal- 
culus. 

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xiv  +  404  pages,  70  figures.     Cloth,  $3.00. 


THE 


THEORY  OF  ERRORS 


AND 


METHOD  OF  LEAST  SQUARES 


BY 


WILLIAM    WOOLSEY  JOHNSON 

PROFESSOR    OF   MATHEMATICS    AT   THE    UNITED -STATES   NAVAL  ACADEMY 
ANNAPOLIS   MARYLAND 


FIRST   EDITION 
SECOND    THOUSAND 


NEW  YORK: 

JOHN  WILEY  &  SONS. 

LONDON  :  CHAPMAN  &  HALL,  LIMITED. 

I9°5 


COPYRIGHT,  1892, 

BY 
W.  WOOLSEY  JOHNSON. 


ASTRONtttff 


ROBERT  DRUMMOND, 

Eifctrotyper, 

444  &  446  Pearl  Street, 

New  York. 


PREFACE. 


THE  basis  adopted  in  this  book  for  the  theory  of  accidental 
errors  is  that  laid  down  by  Gauss  in  the  Theoria  Motus  Cor- 
porum  Coslestium  (republished  as  vol.  vii  of  the  Werke),  which 
may  be  described  for  the  most  part  in  his  own  words,  as  fol- 
lows : 

"  The  hypothesis  is  in  fact  wont  to  be  considered  as  an 
axiom  that,  if  any  quantity  has  been  determined  by  several 
direct  observations,  made  under  similar  circumstances  and 
with  equal  care,  the  arithmetical  mean  between  all  the  observed 
values  presents  the  .most  probable  value,  if  not  with  absolute 
rigor,  at  least  very^nearly  so,  s'6  that  it  is  always  safest  to  ad- 
here to  it."  (Art.  177.) 

Then  introducing  the  notion  of  a  law  of  facility  of  error  to 
give  precise  meaning  to  the  phrase  "  most  probable  value,"  we 
cannot  do  better  than  to  adopt  that  law  of  facility  in  accord- 
ance with  which  the  arithmetical  mean  is  the  most  probable 
value.  After  deriving  this  law  and  showing  that  it  leads  to 
the  principle  of  least  squares,  he  says  :  "  This  principle,  which 
in  all  applications  of  mathematics  to  natural  philosophy  ad- 
mits of  very  frequent  use,  ought  everywhere  to  hold  good  as 
an  axiom  by  the  same  right  as  that  by  which  the  arithmetical 
mean  between  several  observed  values  of  the  same  quantity  is 
adopted  as  the  most  probable  value."  (Art.  179.) 


M298740 


IV  PREFACE. 


Accordingly  no  attempt  has  been  made  to  demonstrate  the 
principle  of  the  arithmetical  mean,  nor  to  establish  the  expo- 
nential law  of  facility  by  any  independent  method.  It  has 
been  deemed  important,  however,  to  show  the  self-consistent 
nature  of  the  law,  in  the  fact  that  its  assumption  for  the  errors 
of  direct  observation  involves  as  a  consequence  a  law  of  the 
same  form  for  any  linear  function  of  observed  quantities,  and 
particularly  for  the  final  determination  which  results  from  our 
method.  This  persistence  in  the  form  of  the  law  has  too 
frequently  been  assumed,  in  order  to  simplify  the  demonstra- 
tions ;  but  at  the  expense  of  soundness. 

No  place  has  been  given  to  the  so-called  criteria  for  the 
rejection  of  doubtful  observations.  Any  doubt  which  attaches 
to  an  observation  on  account  of  the  circumstances  under 
which  it  is  made,  is  recognized,  in  the  practice  of  skilled  ob- 
servers, in  its  rejection,  or  in  assigning  it  a  small  weight  at  the 
time  it  is  made  ;  but  these  criteria  profess  to  justify  the  sub- 
sequent rejection  of  an  observation  on  the  ground  that  its 
residual  is  found  to  exceed  a  certain  limit.  With  respect  to 
this  Professor  Asaph  Hall  says:  "When  observations  have 
been  honestly  made  I  dislike  to  enter  upon  the  process  of  cull- 
ing them.  By  rejecting  the  large  residuals  the  work  is  made 
to  appear  more  accurate  than  it  really  is,  and  thus  we  fail  to 
get  the  right  estimate  of  its  quality."  (The  Orbit  of  lapetus, 
p.  40,  Washington  Observations  for  1882,  Appendix  /.) 

The  notion  that  we  are  entitled  to  reject  an  observation, 
that  is,  to  give  it  no  weight,  when  its  residual  exceeds  a  certain 
limit,  would  seem  to  imply  that  we  ought  to  give  less  than  the 
usual  weight  to  those  observations  whose  residuals  fall  just 
short  of  this  limit,  in  tact  that  we  ought  to  revise  the  obser- 
vations, assigning  weights  which  diminish  as  the  residuals 
increase.  Such  a  process  might  appear  at  first  sight  plausible, 


PREFACE. 


but  it  would  be  equivalent  to  a  complete  departure  from  the 
principle  of  the  arithmetical  mean  and  the  adoption  of  a  new 
law  of  facility.  For  this  we  have  no  justification,  either  from 
theory  or  from  the  examination  of  the  errors  of  extended  sets 
of  observations. 

In  the  discussion  of  Gauss's  method  of  solving  the 
normal  equations,  the  notion  of  the  *  reduced  observation 
equations'  (see  Arts.  154,  155)  which  gives  a  new  interpreta- 
tion to  the  '  reduced  normal  equations '  has  been  introduced 
with  advantage.  This  conception,  although  implied  in 
Gauss's  elegant  discussion  of  the  sum  of  the  squares  of  the 
errors  (see  Art.  160),  seems  not  to  have  appeared  explicitly  in 
any  treatise  prior  to  the  third  edition  of  W.  Jordan's  Handbuch 
der  Vermessungskunde  (Stuttgart,  1888).  To  this  very  complete 
work,  and  to  Oppolzer's  Lehrbuch  zur  Bahnbestimmung  der 
Kometen  und  Plancten,  I  am  indebted  for  the  forms  recom- 
mended for  the  computations  connected  with  Gauss's  method, 
and  for  many  of  the  examples. 

W.  W.  J. 
U.  S.  NAVAL  ACADEMY,  June,  1892. 


CONTENTS. 

i. 

INTRODUCTORY. 

PAGE 

Errors  of  Observation , i 

Objects  of  the  Theory 2 

II. 
INDEPENDENT  OBSERVATIONS  OF  A  SINGLE  QUANTITY. 

The  Arithmetical  Mean 4 

Residuals 4 

Weights 5 

The  Probable  Value        6 

Examples 7 

III. 

PRINCIPLES  OF  PROBABILITY. 

The  Measure  of  Probability 9 

Compound  Events 9 

Repeated  Trials 10 

The  Probability  of  Values  belonging  to  a  Continuous  Series      ...  11 

Curves  of  Probability 12 

Mean  Values  under  a  given  Law  of  Probability 14 

The  Probability  of  Unknown  Hypotheses 16 

Examples 19 

IV. 
THE  LAW  OF  PROBABILITY  OF  ACCIDENTAL  ERRORS. 

The  Facility  of  Errors 21 

The  Probability  of  an  Error  between  given  Limits 23 

The  Probability  of  a  System  of  Observed  Values 24 


viii  CONTENTS. 

,  PAGE 

The  most  Probable  Value  derivable  from  a  given  System  of  Observed 

Values 24 

The  Form  of  the  Facility  Function  corresponding  to  the  Arithmet- 
ical Mean 25 

The  Determination  of  the  Value  of  C 26 

The  Principle  of  Least  Squares 28 

The  Probability  Integral 29 

The  Measure  of  Precision 30 

The  Probable  Error 32 

The  Mean  Absolute  Error 32 

The  Mean  Error 33 

Measures  of  the  Risk  of  Error 34 

Tables  of  the  Probability  Integral  and  Error  Function 36 

Comparison  of  the  Theoretical  and  Actual  Frequency  of  Errors     .     .  37 

The  Distribution  of  Errors  on  a  Plane  Area 38 

Sir  John  Herschel's  Proof  of  the  Law  of  Facility  (foot-note)      ...  39 

The  Surface  of  Probability 40 

The  Probability  of  Hitting  a  Rectangle 40 

The  Probability  of  Hitting  a  Circle 42 

The  Radius  of  the  Probable  Circle 42 

The  most  Probable  Distance 43 

Measures  of  the  Accuracy  of  Shooting 44 

Examples 44 

V. 

THE  COMBINATION  OF  OBSERVATIONS  AND  PROBABLE  ACCURACY  OF 
THE  RESULTS. 

The  Probability  of  the  Arithmetical  Mean 48 

The  Combination  of  Observations  of  Unequal  Precision 50 

Weights  and  Measures  of  Precision 51 

The  Probability  of  the  Weighted  Mean 52 

The  most  Probable  Value  of  h  derivable  from  a  System  of  Observa- 
tions        53 

Equality  of  the  Theoretical  and  Observational  Values  of  the  Mean 

Error  in  the  case  of  Observations  of  Equal  Weight 54 

Formulae  for  the  Mean  and  Probable  Errors 55 

The  most  Probable  Value  of  h  in  Target  Practice 57 

The  Computation  of  the  Probable  Error 58 

The  Values  of  h  and  r  derived  from  the  Mean  Absolute  Error  ...  63 

, 66 


CONTENTS.  IX 

VI. 

THE  FACILITY  OF  ERROR  IN  A  FUNCTION  OF  ONE  OR  MORE 
OBSERVED  QUANTITIES. 

PA-B 

The  Linear  Function  of  a  Single  Observed  Quantity 68 

Non-linear  Functions  of  a  Single  Observed  Quantity 69 

The  Facility  of  Error  in  the  Sum  or  Difference  of   two   Observed 

Quantities 70 

The  Linear  Function  of  Several  Observed  Quantities 72 

The  Non-linear  Function  of  Several  Observed  Quantities      .     .     .     .  73 

Examples 74 

VII. 

THE  COMBINATION  OF  INDEPENDENT  DETERMINATIONS  OF  THE 
SAME  QUANTITY. 

The  Distinction  between  Precision  and  Accuracy 76 

Relative  Accidental  and  Systematic  Errors 78 

The  Relative  Weights  of  Independent  Determinations 79 

The  Combination  of  Discordant  Determinations 81 

Formulae  for  Probable  Error  when  n  =  2  (see  foot-note)       ....  83 

Indicated  and  Concealed  Portions  of  ihe  Risk  of  Error 84 

The  Total  Probable  Error  of  a  Determination 86 

The  Ultimate  Limit  of  Accuracy 88 

Examples 89 

VIII. 

INDIRECT  OBSERVATIONS. 

Observation  Equations 91 

The  Reduction  of  Observation  Equations  to  the  Linear  Form    ...  93 

The  Residual  Equations 94 

Observation  Equations  of  Equal  Precision 94 

The  Normal  Equation  for  x 95 

The  System  of  Normal  Equations 97 

Observation  Equations  of  Unequal  Precision 98 

Formation  of  the  Normal  Equations 99 

The  General  Expressions  for  the  Unknown  Quantities 100 

The  Weights  of  the  Unknown  Quantities 101 

The  Determination  of  the  Measure  of  Precision       .......  105 


X  CONTENTS, 

PAGE 

The  Probable  Errors  of  the  Observations  and  Unknown  Quantities  108 

Expressions  for  2v* 109 

Measure  of  the  Independence  of  the  Observation  Equations    .     .     .in 

Empirical  or  Interpolation  Formulae 112 

Conditioned  Observations       ,     . 113 

The  Correlative  Equations 115 

Examples c,      ,  116 

IX. 
GAUSS'S  METHOD  OF  SUBSTITUTION. 

The  Reduced  Normal  Equations 120 

The  Elimination  Equations 122 

The  Reduced  Observation  Equations 123 

Weights  of  the  Two  Quantities  First  Determined 126 

The  Reduced  Expression  for  2v* 127 

The  General  Expression  for  the  Sum  of  the  Squares  of  the  Errors    .   128 

The  Probability  of  a  Given  Value  of  / 133 

The  Auxiliaries  Expressed  in  Determinant  Form        134 

Form  of  the  Calculation  of  the  Auxiliaries 136 

Check  Equations 138 

Numerical  Example 141 

Values  of  the  Unknown  Quantities  from  the  Elimination  Equations  141 

Independent  Values  of  the  Unknown  Quantities 142 

Computation  of  ai,  «2,  etc 144 

The  Weights  of  the  Unknown  Quantities 145 

Computation  of  the  Weights 148 

Examples 149 

VALUES  OF  CONSTANTS 152 

VALUES  OF  THE  PROBABILITY  INTEGRAL. 

Table  I.— Values  of  Pt 153 

Table  II.— Values  of  Pz 154 

Squares,  Cubes,  Square-roots,  and  Cube-roots 155 


THE  THEORY  OF  ERRORS  AND  METHOD 
OF  LEAST  SQUARES. 


I. 

INTRODUCTORY. 

Errors  of  Observation. 

1.  A  quantity  of  which  the  magnitude  is  to  be  determined  is 
either  directly  measured,  or,  as  in  the  more  usual  case,  deduced 
by  calculation  from  quantities  which  are  directly   measured. 
The  result  of  a  direct  measurement  is  called  an  observation. 
Observations  of  the  kind  here  considered  are  thus  of  the  nature 
of  readings  upon  some  scale,  generally  attached  to  an  instru- 
ment of  observation.     The  least  count  of  the  instrument  is  the 
smallest  difference  recognized  in  the  readings  of  the  instrument, 
so  that  every  observation  is  recorded  as  an  integral  multiple  of 
the  least  count. 

2.  Repeated  observations  of  the  same  quantity,  even  when 
made  with  the  same  instrument  and  apparently  under  the  same 
circumstances,  will  nevertheless  differ  materially.     An  increase 
in  the  nicety  of  the  observations,  and  the  precision  of  the  instru- 
ment, may  decrease  the  discrepancies  in  actual  magnitude ;  but 
at  the  same  time,  by  diminishing  the  least  count,  their  numerical 
measures  will  generally  be  increased ;  so  that,  with  the  most 
refined  instruments,  the  discrepancies  may  amount  to  many 
times  the  least  count.     Thus  every  observation  is  subject  to  an 
error,  the  error  being  the  difference  between  the  observed  value 
and  the  true  value ;  an  observed  value  which  exceeds  the  true 
value  is  regarded  as  having  a  positive  error,  and  one  which  falls 
short  of  it  as  having  a  negative  error. 


INTRODUCTORY.  [Art.  3 


3.  An  error  may  be  regarded  as  the  algebraic  sum  of  a  num- 
ber of  elemental  errors  due  to  various  causes.     So  far  as  these 
causes  can  be  ascertained,  their  results  are  not  errors  at  all,  in 
the  sense  in  which  the  term  is  here  used,  and  are  supposed  to 
have  been  removed  by  means  of  proper  corrections.    Systematic 
errors  are  such  as  result  from  unknown  causes  affecting  all  the 
observations  alike.     These  again  are  not  the  subjects  of  the 
"  theory  of  errors,"  which  is  concerned  solely  with  the  acci- 
dental errors  which  produce  the  discrepancies  between  the 
observations. 

Objects  of  the  Theory. 

4.  It  is  obvious  that  when  a  set  of  repeated  observations  of 
the  same  quantity  are  made,  the  discrepancies  between  them 
enable  us  to  judge  of  the  degree  of  accuracy  we  have  attained. 
Speaking  in  general  terms,  of  two  sets  of  observations,  that  is 
the  best  which  exhibits  upon  the  whole  the  smaller  discrepancies. 
It  is  obvious  also  that  from  a  set  of  observations  we  shall  be 
able  to  obtain  a  result  in  which  we  can  have  greater  confidence 
than  in  any  single  observation. 

It  is  one  of  the  objects  of  the  theory  of  errors  to  deduce  from 
a  number  of  discordant  observations  (supposed  to  be  already 
individually  corrected,  so  far  as  possible)  the  best  attainable 
result,  together  with  a  measure  of  its  accuracy ;  that  is  to  say, 
of  the  degree  of  confidence  we  are  entitled  to  place  in  it. 

5.  When  a  number  of  unknown  quantities  are  to  be  deter- 
mined by  means  of  equations  involving  observed  quantities,  the 
quantities  sought  are  said  to  be  indirectly  observed.    It  is  neces- 
sary to  have  as  many  such  observation  equations  as  there  are 
unknown  quantities.     The  case  considered  is  that  in  which  it  is 
i  n possible  to  make  repeated   observations   of  the  individual 
observed  elements  of  the  equations.     These  may,  for  example, 
be  altitudes  or  other  astronomical  magnitudes  which  vary  with 
the  time,  so  that  the  corresponding  times  are  also  among  the 
observed  quantities.    Nevertheless,  there  is  the  same  advantage 
in  employing  a  large  number  of  observation  equations  that  there 


§1.]  OBJECTS  OF  THE   THEORY.  3 

is  in  the  repetition  of  direct  observations  upon  a  single  required 
quantity.  If  there  are  n  unknown  quantities,  any  group  con- 
taining n  of  the  equations  would  determine  a  set  of  values  for 
the  unknown  quantities ;  but  these  values  would  differ  from 
those  given  by  any  other  group  of  n  of  the  equations. 

We  may  now  state  more  generally  the  object  of  the  theory  of 
errors  to  be,  when  given  more  than  n  observation  equations 
involving  n  unknown  quantities,  the  equations  being  somewhat 
inconsistent,  to  derive  from  them  the  best  determination  of  the 
values  of  the  several  unknown  quantities,  together  with  a 
measure  of  the  degree  of  accuracy  attained. 

6.  It  will  be  noticed  that,  putting  n  —  i,  this  general  state- 
ment includes  the  case  of  direct  observations,  in  which  all  the 
equations  are  of  the  form 

X  =  xl ,     X  =  Xi ,    .  .  •  , 

where  X  is  the  quantity  to  be  determined,  and  each  equation 
gives  an  independent  statement  of  its  value. 

We  commence  with  this  case  of  direct  observations  of  a  single 
quantity,  and  our  first  consideration  will  be  that  of  the  best 
determination  which  can  be  obtained  from  a  number  of  such 
observations. 


n. 

INDEPENDENT  OBSERVATIONS  OF  A  SINGLE  QUANTITY. 

The  Arithmetical  Mean. 

7.  Whatever  rule  we  adopt  for  deducing  the  value  to  be 
accepted  as  the  final  result  derived  from  several  independent 
observations,  it  must  obviously  be  such  that  when  the  observa- 
tions are  equal  the  result  shall  be  the  same  as  their  common 
value.  When  the  observations  are  discordant,  such  a  rule  pro- 
duces an  intermediate  or  mean  value.  Thus,  if  there  be  n 
quantities,  x± ,  x^ ,  .  .  .  xn ,  the  expressions 

IX  n..  IIX1 

— ,     V  (•*!•*••  •  ••*.»),     y  — >     etc., 

give  different  sorts  of  mean  values.  Of  these,  the  one  first 
written,  which  is  the  arithmetical  mean,  is  the  simplest,  and  it 
is  also  that  which  has  universally  been  accepted  as  the  final 
value  when  xlt  x^  . .  .  xn  are  independently  observed  values  of 
a  single  quantity  x,  the  observations  being  all  supposed  equally 
good. 

Residuals. 

8.  The  differences  between  the  several  observed  values  and 
the  value  which  we  take  as  our  final  determination  of  the  true 
value  are  called  the  residuals  of  the  observations.  The  resid  uals 
are  then  what  we  take  to  be  the  errors  of  the  observations ;  but 
they  differ  from  them,cf  course,  by  the  amount  of  error  existing 
in  our  final  determination.  If  the  observed  values  were  laid 
down  upon  a  straight  line,  as  measured  from  any  origin,  the 
residuals  would  be  the  abscissas  of  the  points  thus  representing 
the  observations  when  the  point  corresponding  to  the  final  value 
adopted  is  taken  as  the  origin. 


§  II.]  RESIDUALS.  5 

9.  In  the  case  of  the  arithmetical  mean,  the  algebraic  sum  of 
the  residuals  is  zero.     For,  if  a  denote  the  arithmetical  mean  of 
the  n  quantities  xl ,  x^ ,  .  .  .  xn ,  we  have 

•-¥ <•> 

the  residuals  are 

Xi  —  a,  >  Xi  —  a,    ...    xn  —  a, 

and  their  sum  is 

Ix  -  na , 

which  is  zero  by  equation  (i). 

When  the  observations  are  represented  by  points,  as  in  the 
preceding  article,  the  geometrical  mean  point  or  centre  of 
gravity  of  these  points  is  the  point  whose  abscissa  is  a,  and, 
when  this  point  is  taken  as  the  origin,  the  sum  of  the  positive 
abscissas  of  observation  points  is  equal  to  the  sum  of  the  nega- 
tive abscissas. 

Weights. 

10.  When  the  observations  are  not  made  under  the  same  cir- 
cumstances, and  are  therefore  not  regarded  as  equally  good,  a 
greater  relative  importance  can  be  given  to  a  better  observation 
by  treating  it  as  equivalent  to  more  than  one  occurrence  of  the 
same  observed  value  in  a  set  of  equally  good  observations. 
For  example,  if  there  were  two  observations  giving  the  observed 
values  xv  and  x^ ,  and  the  first  observation  were  regarded  as  the 
best,  we  might  proceed  as  if  the  observed  value  x^  occurred 
twice  and  xt  once  in  a  set  of  three  observations  equally  good. 
The  arithmetical  mean  would  then  be 


In  this  process  we  are  said  to  give  to  the  observations  the  rela- 
tive weights  of  2  and  i.  The  weight  may  be  regarded  as  the 
numerical  measure  of  the  influence  of  the  observation  upon  the 
arithmetical  mean. 


6  OBSERVATIONS  OF  A  SINGLE  QUANTITY.      [Art.  11 

11.  In  general,  A, /»2 ,  .  .  . pn  being  taken  as  the  weights  of 
the  observations  xly  xz ,  .  .  .  xn ,  the  arithmetical  mean  with 
these  weights  is 

a  = 

This  expression  is  called  the  weighted  arithmetical  mean. 
When  the  weights  are  integers,  it  is  the  same  as  the  arithmetical 
mean  of  Zp  observations,  of  which  pl  give  the  observed  value 
Xi ,  pi,  the  observed  value  xz ,  and  so  on.  But,  since  only  the 
ratios  of  the  weights  affect  the  result,  it  is  not  necessary  to 
suppose  them  to  be  integers. 

It  is  easily  shown,  as  in  Art.  9,  that,  if  the  residuals  are  mul- 
tiplied by  the  weights,  the  algebraic  sum  of  the  results  is  zero. 
Again,  when  as  in  that  article  the  observations  are  represented 
by  points,  the  point  whose  abscissa  is  the  weighted  mean  is  the 
centre  of  gravity  of  bodies  placed  at  the  observation  points 
having  weights  proportional  top^p*,  .  .  .  pn* 

12.  The  weight  of  a  result  obtained  by  the  rule  given  above 
is  defined  to  be  the  sum  of  the  weights  of  its  constituents ;  so 
that,  because 

alp  =  Ipx, 

the  product  of  a  result  by  its  weight  is  equal  to  the  sum  of  the 
like  products  for  its  constituents.  It  follows  that,  in  obtaining 
the  final  result,  we  may  for  any  group  of  observations  substitute 
their  mean  with  the  proper  weight. 

In  the  case  of  observations  supposed  equally  good,  the  weight 
of  each  is  taken  equal  to  unity,  and  then  the  weight  of  the  mean 
is  the  number  of  observations. 

The  Probable  Value. 

13.  The  most  probable  value  of  the  observed  quantity,  or 
sirrfply  the  probable  value,  in  the  ordinary  sense  of  the  expres- 
sion signifies  that  which,  in  our  actual  state  of  knowledge,  we 
are  justified  in  considering  as  more  likely  than  any  other  to  be 
the  true  value.     In  this  sense,  the  arithmetical  mean  is  the  most 


§11.]  THE  PROBABLE   VALUE.  7 

probable  value  which  can  be  derived  from  observations  con- 
sidered equally  good.  This  is,  in  fact,  equivalent  to  saying 
that  we  accept  the  arithmetical  mean  as  the  best  rule  for  com- 
bining the  observations,  having  no  reason  either  theoretical  or 
practical  for  preferring  any  other.* 

But,  if  instead  of  a  rule  of  combination  we  adopt  a  theory 
with  respect  to  the  nature  of  accidental  errors,  the  probable 
value  will  depend  upon  the  adopted  theory.  To  become  the 
subject  of  mathematical  treatment  such  a  theory  must  take  the 
shape  of  a  law  of  the  probability  of  accidental  errors,  as  will  be 
explained  in  a  subsequent  section.  Since,  in  the  nature  of 
things,  this  law  can  never  be  absolutely  known,  and  since  more- 
over it  probably  differs  with  differing  circumstances  of  observa- 
tion, the  most  probable  value  in  this  technical  sense  is  itself 
unknown.  But  when  the  expression  is  used  without  specifying 
the  law  of  probability,  it  signifies  the  value  which  is  the  most 
probable  in  accordance  with  the  generally  accepted  law  of  proba- 
bility. Before  proceeding  to  this  law,  we  shall  consider,  in  the 
following  section,  the  principles  of  probability  so  far  as  we  shall 
need  to  apply  them. 

Examples. 

1.  Show  that  the  formula  nf(a)  =  2f{x)  determines  a  mean 
value  of  n  quantities  for  any  form  of  the  function/,  and  that  the 
geometric  mean  is  included  in  this  rule. 

2.  Except  when/(.#)  =  ex  in  Ex.  i,  the  position  of  the  point 
whose  abscissa  is  a  is  dependent  upon  the  position  of  the  origin 
as  well  as  upon  the  observation  points. 

*That  the  most  probable  value,  when  there  are  but  two  observations, 
is  their  arithmetical  mean  follows  rigorously  from  the  hypothesis  that 
positive  and  negative  errors  are  equally  probable.  The  property  of  the 
arithmetical  mean  pointed  out  in  Art.  12  shows  that  the  result  for  three 
observations  is  expressible  as  a  function  of  the  result  for  two  of  them 
and  the  third  observation,  and  so  on  for  four  or  more  observations.  It 
was  upon  the  assumption  that  the  most  probable  value  must  possess  this 
property  that  Encke  based  his  so-called  proof  that  the  arithmetical  mean 
is  the  most  probable  value  for  any  number  of  observations  {Berliner 
Astronomisches  Jahrbuch  for  1834,  pp.  260-262). 


8  OBSERVATIONS  OF  A  SINGLE  QUANTITY.      [Art.  13 

3.  If  the  values  of  x  are  nearly  equal  in  Ex.  i,  the  result  of  the 
formula  is  nearly  equivalent  to  a  weighted  arithmetical  mean  in 
which  the  weights  are  proportional  tof'(lxl  +  i«),  f'(^x^  +  \a)t 
etc. 

4.  When  a  mean  value  is  determined  by  an  equation  of  the 
form  2f(x  —  a)  =  o,  the  position  of  the  point  whose  abscissa  is  a 
is  independent  of  the  origin.    Give  the  cubic  determining  a  when 
2(x  —  0)3  =  o,  and  show  that  one  root  only  is  real. 

5.  Prove  that  the  weighted  arithmetical  mean  of  values  of 
x  +  y  is  the  sum  of  the  like  means  of  the  values  of  x  and  of  the 
values  of  y  respectively. 


in. 

PRINCIPLES  OF  PROBABILITY. 

The  Measure  of  Probability. 

14.  'Tte  probability  Q{  a.  future  event  is  the  measure  of  our 
reasonable  expectation  of  the  event  in  our  present  state  of 
knowledge  of  its  causes.  Thus,  not  knowing  any  reason  to  the 
contrary,  when  a  die  is  to  be  thrown  we  assign  an  equal  proba- 
bility to  the  several  events  of  the  turning  up  of  its  six  different 
faces.  We  say,  therefore,  that  the  probability  or  chance  that 
the  ace  will  turn  up  is  i  to  5,  or  better,  i  out  of  6,  hence  the 
fraction  \  is  taken  as  the  measure  of  the  probability.  Thus  the 
probability  of  an  event  which  is  one  of  a  set  of  equally  likely 
events,  one  of  which  must  happen,  is  the  fraction  whose  num- 
erator is  unity  and  whose  denominator  is  the  number  of  these 
events.  Obviously,  the  probability  of  an  event  which  can  happen 
in  several  ways  is  the  sum  of  the  probabilities  of  the  sever  always. 
Thus  if  the  die  had  two  blank  faces,  the  probability  that  one  of 
them  would  turn  up  would  be  -|  or  \.  The  sum  of  the  proba- 
bilities of  all  the  possible  events  is  unity,  which  represents  the 
certainty  that  some  one  of  the  events  will  happen. 

Compound  Events. 

15.  An  event  which  consists  of  the  joint  occurrence  of  two 
independent  events  is  called  a  compound  event.  By  independent 
events  we  mean  events  such  that  the  occurrence  or  non-occur- 
rence of  the  first  has  no  influence  upon  the  occurrence  or  non- 
occurrence  of  the  second.  For  example,  the  throwing  of  sixes 
with  a  pair  of  dice  is  a  compound  event  consisting  of  the  turning 
up  of  a  special  face  of  each  die.  The  whole  number  of  com- 
pound events  is  evidently  the  product  of  the  numbers  of  simple 
events ;  and,  since  the  several  probabilities  are  the  reciprocals 


10  PRINCIPLES  OF  PROBABILITY.  [Art.  15 

of  these  numbers,  the  probability  of  the  compound  event  is  the 
product  of  the  probabilities  of  the  simple  events.  Thus,  when  a 
pair  of  dice  is  thrown  we  have  6  X  6  =  36  compound  events, 
and  the  probability  of  a  special  one,  such  as  the  throwing  of 
sixes,  is  £  X  £  =  A- 

In  like  manner,  if  more  than  two  simple  events  are  concerned, 
it  is  easily  seen  that,  in  general,  the  probability  of  a  compound 
event  is  the  product  of  the  probabilities  of  the  independent  simple 
events  of  whose  joint  occurrence  it  consists. 

16.  A  compound  event  may  happen  in  different  ways,  and 
then,  of  course,  the  probabilities  of  these  independent  ways  must 
be  added.     For  example,  six  and  five  may  be  thrown  in  two 
ways,  that  is  to  say,  two  of  the  36  equally  likely  events  consist 
of  the  combination  six  and  five,  hence  the  chance  is  -^  or  ^ 
A  throw  whose  sum  amounts  to  10  can  occur  in  three  ways, 
therefore  its  chance  is  -£$  or  -J^. 

Repeated  Trials. 

17.  When  repeated  opportunities  for  the  occurrence  or  non- 
occurrence  of  the  same  set  of  events  can  be  made  to  take  place 
under  exactly  the  same  circumstances,  equally  probable  events 
will  tend  to  occur  with  the  same  frequency.     Therefore,  in  a 
large  number  of  such  opportunities  or  trials,  the  relative  fre- 
quency of  the  occurrence  of  an  event  which  can  happen  in  m 
ways  and  fail  in  n  ways  (the  m  +  n  ways  of  both  kinds  corres- 
ponding to  m  +  n  equally  probable  elementary  events)  will  tend 

to  the  value  — m— ,  which  is  the  fraction  expressing  the  prob- 
m  +  n 

ability  of  the  event.  This  is  commonly  expressed  by  saying 
that  the  ratio  of  the  number  of  occurrences  of  an  event  to  the 
whole  number  of  trials  will  "m  the  long  run"  be  the  fraction 
which  expresses  the  probability.  The  correspondence  of  this 
frequency  in  the  long  run  with  the  estimated  probability  forms 
the  only  mode,  though  an  uncertain  one,  of  submitting  our 
results  to  the  test  of  experience. 


§m.] 


PROBABILITY  OF  CONTINUOUS  VALUES. 


II 


The  Probability  of  Values  belonging  to  a  Continuous  Series. 

1 8.  In  the  examples  given  in  the  preceding  articles,  the  equally 
probable  elementary  events,  which  are  the  basis  of  our  estimate 
of  probability,  form  a  limited  number  of  distinct  events,  such  as 
the  turning  up  of  the  different  faces  of  a  die.  But,  in  many 
applications,  these  events  belong  to  a  consecutive  series,  inca- 
pable of  numeration.  For  example,  suppose  we  are  concerned 
with  the  value  of  a  quantity  x>  of  which  it  is  known  that  any 
value  between  certain  limits  a  and  b  is  possible ;  or,  what  is  the 
same  thing,  the  position  of  the  point  P,  whose  abscissa  is  x,  when 
P  may  have  any  position  between  certain  extreme  points  A 
and  B.  We  cannot  now  assign  any  finite  measure  to  the  prob- 
ability that  x  shall  have  a  definite  value,  or  that  P  shall  fall  at  a 
definite  point,  because  the  number  of  points  upon  the  line  AB 
is  unlimited.  We  have  rather  to  consider  the  probability  that 
P  shall  fall  upon  a  definite  segment  of  the  line,  or  that  the  value 
of  x  shall  lie  between  certain  limits. 

IQ.  It  is  customary,  however,  to  compare  the  probabilities 
that  P  shall  fall  at  certain  points.  Suppose  in  the  first  place 

C  D 


& 

i 
i 

i 

J* 

B 


FIG.  i. 


that,  when  any  equal  segments  of  the  line  AB  are  taken,  the 
probabilities  that  P  shall  fall  in  these  segments  are  equal.  In 
this  case,  the  probability  that  P  shall  fall  at  a  given  point  is  said 
to  be  constant  for  all  points  of  the  line.  Let  Ax  be  a  segment 
of  the  line  AB\  then,  if  the  probability  for  all  points  of  AB  is 
constant,  it  readily  follows  from  the  definition  just  given  that  the 


12  PRINCIPLES  OF  PROBABILITY.  [Art.  ig 

probability  that  P  shall  fall  in  the  segment  Ax  is  proportional 
to  Ax.  Since  we  suppose  it  certain  that  P  shall  fall  somewhere 
between  A  and  B,  this  probability  will  be  represented  by 

Ax  Ax 

or 


AB  b-a 

Let  an  ordinate  y  be  taken  such  that  y  Ax  is  the  value  of  this 
probability  ;  then 


and,  constructing  as  in  Fig.  i  the  line  CD  having  this  constant 
ordinate,  the  probabilities  for  any  segments  of  AB  are  the  cor- 
responding rectangles  contained  between  the  axis  and  the  line 
CD  For  different  values  of  the  limiting  space  AB  in  which  P 
may  fall,_y  varies  in  inverse  ratio.  Thus,  if  AB  is  changed  to 
AB*  ',  the  new  ordinate  AC'  or  y'  is  such  that  yf.AB'=y.AB, 
each  of  the  areas  A  CDB  and  A  C'D'B'  being  equal  to  unity. 
The  two  values  of  y  are  said  to  determine  the  relative  proba- 
bilities that  P  shall  fall  at  a  given  point  in  the  two  cases. 

Curves  of  Probability. 

20.  Taking  now  the  case  in  which  the  probability  is  not  con- 
stant for  all  points,  let  AB  be  divided  into  segments,  and  let 
rectangles  be  erected  upon  them,  the  area  of  each  rectangle 
representing  the  probability  that  /'shall  fall  in  the  corresponding 
segment.    The  heights  of  these  rectangles  will  now  differ  for  the 
different  segments.     Denoting  the  height  for  a  given  segment 
Ax  by  y,  the  relative  values  of  y  for  any  two  segments  deter- 
mme,  as  explained  in  the  preceding  article,  the  relative  proba- 
bility that  P  shall  fall  at  a  given  point  in  one  or  the  other  of  the 
segments,  on  the  hypothesis  that  the  probability  is   constant 
throughout  the  segment.     They  may  thus  be  said  to  measure 
the  mean  values  of  the  probabilities  for  given  points  taken  in  the 
various  segments.     The  sum  of  the  areas  of  the  rectangles  will, 
of  course  be  unity  ;  that  is.  IyAx=  i. 

21.  If  we  now  subdivide  the  segments,  the  figure  composed 


III.] 


CURVES  OF  PROBABILITY. 


of  the  sum  of  the  rectangles  will  approach  more  and  more  nearly, 
as  we  diminish  the  segments  without  limit,  to  a  curvilinear  area, 
and  the  variable  ordinate  of  the  limiting  curve  will  measure  the 
continuously  varying  probability  that  P  shall  fall  at  a  given  point 
of  the  line  AB. 

The  value  ofj>  is  now  a  continuous  function  of  x  the  abscissa 
of  the  corresponding  point,  and,  putting^/  =f(x),  the  function 
f(x)  is  said  to  express  the  law  of  the  probability  of  the  value  x. 


A  B 

FIG.  2. 

The  curve  y  =/(.#)  is  the  probability  curve  corresponding 
to  the  given  la.wf(x).  The  entire  area  ACDB,  Fig.  2,  whose 

(6 
ydx  (which  is  the  limit  of  2yAx ;  see  Int.  Calc.,  Art.  99), 
a 

a  and  b  being  the  limiting  values  between  which  x  certainly  falls, 
is  equal  to  unity.  In  general,  for  any  limits  the  value  of  the 

IP 

integral    ydx  is  the  probability  that  x  falls  between  the  values 

«  and  /?.  The  element  ydx  of  this  integral  may  be  called  the 
element  of  probability  for  the  value  x.  It  is  sometimes  called 
the  probability  that  the  value  shall  fall  between  x  and  x  +  dx, 
it  being  in  that  case  understood  that  dx  is  taken  so  small  that 
the  probability  may  be  regarded  as  constant  in  this  interval. 

22.  As  an  illustration  of  what  precedes,  suppose  it  to  be 
known  that  the  value  of  x  must  fall  between  zero  and  a,  and  that 
the  probabilities  of  values  between  these  limits  are  proportional 
to  the  values  themselves.  These  conditions  give 


and 


y 

Ia 
ydx  =  i , 
o 


14  PRINCIPLES  OF  PROBABILITY.  [Art.  22 

whence,  substituting  and  integrating, 

ca*  2 

T  =  .,    or    ,=  -. 

Hence  the  law  of  probability  in  this  case  is 

2X 


We  may  now  find  the  probability  that  x  shall  fall  between  any 
given  limits.  For  example,  the  probability  that  x  shall  exceed 
\a  is  represented  by 

fa  o 


4 

Thus  the  odds  are  3  to  i  that  x  exceeds  \a  when  the  law  of 
probability  is  that  proposed. 

Mean  Values  under  a  given  Law  of  Probability. 

23.  When  a  quantity  x  has  a  given  law  of  probability,  we 
have  frequently  occasion  to  consider  what  would  be  its  mean  or 
average  value  "  in  the  long  run,"  that  is  to  say,  the  arithmetical 
mean  of  its  values,  supposing  them  to  occur  in  a  large  number 
of  trials  with  the  frequency  indicated  by  the  given  law  of  prob- 
ability. See  Art.  17. 

Let  us  suppose,  in  the  first  place,  that  only  a  limited  number 
of  distinct  values,  say 

are  possible.  Let  /\ ,  P9 .  .  .  Pm  be  the  proper  fractions  which 
represent  the  respective  probabilities  of  these  values.  Then,  in 
a  large  number  n  of  trials,  the  number  of  times  in  which  the 
distinct  values  x± ,  xz .  .  .  xm  occur  will  be 

#/>! ,  nP2 ,  .  .  .  nPm 

respectively.  The  arithmetical  mean  mentioned  above  is,  there- 
fore, 

+  nP*Xi  +  .  .  .  +  nPmxm 


§111.]  MEAN  VALUES  UNDER  A   GIVEN  LAW. 


that  is,  PI#I  +  P^x2  +  .  .  .  +  PmXm  y 

or  -Px . 

That  is  to  say,  the  mean  value  is  found  by  multiplying  the  m 
distinct  values  by  their  probabilities  and  adding  the  results.* 
24.  Next,  supposing  a  continuous  series  of  values  possible, 
let  y  Ax  be  taken,  as  in  Art.  20,  to  represent  the  probability  that 
x  falls  between  x  and  x  +  Ax.  Evidently,  in  each  term  of 
IPx ,  we  must  now  substitute  this  expression  for  P,  and  for  x 
some  intermediate  value  between  x  and  x  +  Ax.  When  we 
pass  to  the  limit,  in  which  y  becomes  a  continuous  function  of  xy 
this  sum  becomes 

(b 
xydx, 
a 

which  is  thus  the  mean  value  of  x,  when  y  is  the  function 
expressing  its  law  of  probability  and  a  and  b  its  extreme 
possible  values. 

For  example,  with  the  law  of  probability  considered  in  Art. 
22,  namely, 

2X 


the  mean  value  of  x  is 


25.  In  the  same  manner  it  may  be  shown  that,  if  y=f(x) 
expresses  the  law  of  probability  of  x,  the  mean  value  of  any 
function  F(x)  is 

?  F(x)f(x)dx. 

Ja 

*  The  "value  of  an  expectation"  is  an  instance  of  a  mean  value. 
Thus,  if  x1  is  the  value  to  be  received  in  case  a  certain  event  whose  prob- 
ability is  PI  happens,  x^  the  value  to  be  received  if  an  event  whose 
probability  is  P2  happens,  and  so  on  for  m  distinct  events,  one  of  which 
must  happen,  then  the  mean  value  ZPx  is  called  the  value  of  the  expec- 
tation. 


1 6  PRINCIPLES  OF  PROBABILITY.  [Art.  25 

Thus,  again  taking  the  law  of  probability  y  =  — ,  the  mean 

value  of  x*  *  is 

c? 

2 

Again,  that  of  —  is 


26.  If  all  values  between  a  and  b  are  equally  probable,  the 
ement  of 
this  case,  is 


element  of  probability  is  -f  -  ;  thus  the  mean  value  of  x,  in 


xdx 


ab-a~2(b-a)~       2     ' 

which  is  the  same  as  the  arithmetical  mean  between  the  limiting 
values.     Again,  the  mean  value  of  x*t  in  this  case,  is 

^ 


The  Probability  of  Unknown  Hypotheses. 

27.  No  distinction  can  be  drawn  between  the  probability  of 
an  uncertain  future  event  and  that  of  an  unknown  contingency,  in 
a  case  where  the  decisive  "event"  has  indeed  happened,  but  we 
remain  in  doubt  with  regard  to  it  because  only  probable  evidence 

*  It  should  be  noticed  that  if  z  =  F(x],  the  law  of  probability  for  z  is  not 
found  by  simply  express-ing/^*)  as  a  function  of  z.  It  is  necessary  to 
transform  the  element  of  probability  f(x)afxt  which  expresses  the  proba- 
bility that  x  falls  between  x  and  x  -\-  dx,  and  therefore  represents  also 
the  probability  that  z  falls  between  z  and  z  +  dz.  Thus,  in  the  present 
case,  putting  z  =  x2, 

f(x}dx  =  2jr  dx  =  -jr  , 

which  indicates  that  all  values  of  z  between  o  and  a*  are  equally  prob- 
able when,  as  supposed  in  Art.  22,  the  probability  of  a  value  of  x  is  pro- 
portional to  the  value  itself. 


§IIL]      PROBABILITY  OF  UNKNOWN  HYPOTHESES.          \J 

is  known  to  us.  In  any  case,  the  probability  is  a  mental  estimate 
of  credibility  depending  only  upon  the  known  data,  and  there- 
fore subject  to  change  whenever  new  evidence  becomes  known. 
Let  there  be  two  hypotheses  A  and  B,  one  of  which  must  be 
true,  and  which  so  far  as  we  know  are  equally  probable,  and 
suppose  that  a  trial  is  to  be  made  which  on  either  hypothesis 
may  eventuate  in  one  or  the  other  of  two  ways ;  in  other  words, 
that  an  event  X  may  or  may  not  happen.  Suppose,  further, 
that  on  the  hypothesis  A  the  probability  of  X  is  a,  and  on  the 
hypothesis  B  the  probability  of  X  is  b.  Now  it  is  clear  that 
after  the  trial  has  been  made  and  the  event  X  has  happened, 
we  are  entitled  to  make  a  different  estimate  of  the  relative 
credibilities  of  the  hypotheses  A  and  B. 

28.  To  obtain  the  new  measures  of  the  probabilities  of  A  and 
B,  we  employ  the  notion  of  relative  frequency  in  the  long  run. 
Let  us  then  consider  a  great  number  of  cases  of  the  four  kinds 
which  before  the  event  Jf  we  regard  as  possible,  the  frequencies 
of  the  different  kinds  being  proportional  to  their  probabilities 
as  we  estimate  them  before  the  event.  The  hypotheses  A  and  B 
respectively  are  true  in  an  equal  number  of  cases,  say  n,  of  each. 
The  event  X  will  happen  in  na  of  the  cases  in  which  A  is  true, 
and  not  happen  in  n  (i  —  a)  cases.  Again,  X  will  happen  in  nb 
cases  in  which  B  is  the  true  hypothesis,  and  not  happen  in 
n(i  —  £)  cases. 

Now,  since  Jfhas  actually  happened,  from  the  whole  number, 
2n,  of  cases  we  must  exclude  those  in  which  X  does  not  happen, 
and  consider  only  the  na  +  nb  cases  in  which  X  does  happen. 

Attending  only  to  these  cases,  the  relative  frequency  of  those  in 
which  A  and  B  respectively  are  true  is  the  measure  of  our  present 
estimate  of  their  relative  probability.  Hence  these  probabilities 

are  in  the  ratio  a :  bt  that  is,  the  probability  of  A  is          , ,  and 
/  a  +  o 

that  of  B  is  — ~. 
a  +  b 

2Q.  As  an  illustration,  suppose  there  are  two  bags,  A  and  B, 
containing  white  and  black  balls,  A  containing  3  white  and  5 


1 8  PRINCIPLES  OF  PROBABILITY.  [Art.  29 

black  balls,  B  containing  5  white  and  i  black  ball.  One  of  the 
bags  is  chosen  at  random,  and  then  a  ball  is  drawn  at  random 
from  the  bag  chosen.  The  ball  is  found  to  be  white ;  what  is 
the  probability  that  the  bag  A  was  chosen  ?  Here  a  =  f ,  since 
three  out  of  eight  balls  in  A  are  white,  and  b  =  -f ;  hence  the 
probabilities  are  in  the  ratio  f  :  f  or  9 :  20.  The  probability  that 
the  bag  was  A  is  therefore  -^-. 

Again,  suppose  A  is  known  to  contain  only  white  balls,  and 
B  an  equal  number  of  white  and  black.  If  a  white  ball  is  drawn 
a  =  i,  b  =  \,  the  odds  in  favor  of  A  are  2  :  i  or  the  probability 
of  A  is  f.  But  if  a  black  ball  had  been  drawn,  we  should  have 
had  a  —  o,  b  =  -J-,  the  probability  of  A  is  zero,  that  is,  it  is  certain 
that  the  bag  chosen  was  not  A. 

30.  If  there  are  other  hypotheses  besides  A  and  B  consistent 
with  the  event  X,  the  same  reasoning  as  in  Art.  28  establishes 
the  general  theorem  that  the  probabilities  of  the  several  hypoth- 
eses',  which  before  an  event  X  were  considered  equally  probable* 
are  after  the  event  proportional  to  the  numbers  which  before  the 
event  express  the  probabilities  of  X  on  the  several  hypotheses. 

The  various  hypotheses  in  question  may  consist  in  attributing 
different  values  to  an  unknown  quantity  x,  and  these  values  may 
constitute  a  continuous  series.  The  probabilities  of  the  various 
values  will  then  be  proportional  to  the  corresponding  prob- 
abilities of  the  event  X.  Hence,  to  find  the  law  of  the  prob- 
ability of  x,  it  is  only  necessary  to  determine  a  constant  in  the 
same  manner  that  c  is  determined  in  Art.  22. 

In  particular  it  is  to  be  noticed  that,  of  all  the  values  of  an 
unknown  quantity  which  before  the  occurrence  of  a  certain  event 
were  equally  probable,  that  one  is  after  the  event  the  most  prob- 
able which  before  the  event  assigned  to  it  the  greatest  probability. 

*  If  this  is  not  the  case,  the  probabilities  before  the  event  are  called  the 
antecedent  or  a  priori  probabilities,  and  the  theorem  is  that  the  ratio  of 
the  antecedent  probabilities  is  to  be  multiplied  by  the  probabilities  of  X 
on  the  several  hypotheses,  in  order  to  find  the  ratio  of  the  probabilities 
after  the  event. 


§111.]  EXAMPLES.  19 

Examples. 

1.  From  2n  counters  marked  with  consecutive  numbers  two 
are  drawn  at  random ;  show  that  the  odds  against  an  even  sum 
are  n  to  n  —  i. 

2.  A  and  B  play  chess,  A  wins  on  an  average  2  out  of  3 
games ;  what  is  the  chance  that  A  wins  exactly  4  games  out  of 
the  first  six?  ^. 

3.  A  domino  is  chosen  from  a  set  and  a  pair  of  dice  is  thrown ; 
•what  is  the  chance  that  the  numbers  agree  ?  -£%. 

4.  Show  that  the  chance  of  throwing  9  with  two  dice  is  to  the 
chance  of  throwing  9  with  three  dice  as  24  to  25. 

5.  A  and  B  shoot  alternately  at  a  mark.     A  hits  once  in  n 
times,  B  once  in  n  —  i  times;  show  that  their  chances  of  first  hit 
are  equal,  and  find  the  odds  in  favor  of  B  after  A  has  missed 
the  first  shot.  n  to  n  —  2. 

6.  A  and  B  throw  a  pair  of  dice  in  turn.     A  wins  if  he  throws 
numbers  whose  sum  is  6  before  B  throws  numbers  whose  sum 
is  7 ;  show  that  his  chance  is  |-J. 

7.  A  walks  at  a  rate  known  to  be  between  3  and  4  miles  an 
hour.     He  starts  to  walk  20  miles,  and  B  starts  one  hour  later, 
walking  at  the  rate  of  4  miles  an  hour.     What  is  the  chance  of 
overtaking  him  :  i°  if  all  distances  per  hour  between  the  limits 
are  equally  probable;  2°  if  all  times  per  mile  between  the  limits 
are  equally  probable?  i°,  i  to  2  ;  2°,  2  to  3. 

8.  If  all  values  of  x  between  o  and  a  are  possible  and  their 
probabilities  are  proportional  to  their  squares,  show  that  the 
probability  that  x  exceeds  \a  is  £ ,  and  find  the  mean  value 
of  x.  \a. 

9.  If,  in  the  preceding   example,  we  are  informed   that  x 
exceeds  \a,  how  is  the  probability  affected,  and  what  is  now 
the  mean  value  of  xl  |-f«. 

10.  If  two  points  he  taken  at  random  upon  a  straight  line 
AB,  whose  length  is  a,  and  X  denote  that  which  is  nearest  A, 
show  that  the  curve  of  probability  for  Xis  a  straight  line  passing 
through  B,  and  find  the  mean  value  of  A X.  \a. 


20  PRINCIPLES  OF  PROBABILITY.  [Art.  30 

___  _  *£  ___  ,  _  .  _  .  _  , 

11.  On  a  line  AB,  whose  length  is  a,  a  point  Zis  taken  at 
random,  and  then  a  point  X  is  taken  at  random  upon  AZ. 
Determine  the  probability  curve  for  AX,  or  x,  and  the  mean 
value  of  x.  _i_  j        a  .    a_ 

a       5  x  '   4  ' 

12.  Two  points  are  taken  at  random  on  the  circumference 
of  a  circle  whose  radius  is  a.     Show  that  the  chord  is  as  likely 
as  not  to  exceed  a  V  2,  but  that  the  average  length  of  the  chord 


13.  In  a  semicircle  whose  radius  is  a,  find  the  mean  ordinate  : 
i°  when  all  points  of  the  semi-circumference  are  equally  prob- 
able ;  2°  when  all  points  on  the  diameter  are  equally  probable. 


14.  A  card  is  missing  from  a  pack;  13  cards  are  drawn  at 
random  and  found  to  be  black.     Show  that  it  is  2  to  i  that  the 
missing  card  is  red. 

15.  A  card  has  been  dropped  from  a  pack  ;  13  cards  are  then 
drawn  and  found  to  be  2  spades,   3  clubs,  4  hearts,  and   4 
diamonds.     What,  are  the  relative  probabilities  that  the  missing 
card  belongs  to  the  suits  in  the  order  named  ?       1  1  :  10  :  9  :  9. 

16.  A  and  B  play  at  chess:  when  A  has  the  first  move  the 
odds  are  1  1  to  6  in  favor  of  A,  but  when  B  has  the  first  move 
the  odds  are  only  9  to  5.     A  has  won  a  game;  what  are  the 
odds  that  he  had  the  first  move  ?  154  to  153. 

17.  The  odds  are  2  to  i  that  a  man  will  write  'rigorous' 
rather  than  '  rigourous.'     The  word  has  been  written,  and  a 
letter  taken  at  random  from  it  is  found  to  be  '  u  ';  what  are 
now  the  odds  ?  9  to  8. 

18.  A  point  P  was  taken  at  random  upon  aline  AB,and 
then   a  point   C  was  taken  at  random  upon  AP.     If  we  are 
informed  that  C  is  the  middle  point  of  ABt  what  is  now  the 
probability  curve  of  AP  ?  _       i 

~  X  log  2* 


IV. 

THE  LAW  OF  PROBABILITY  OF  ACCIDENTAL  ERRORS. 
The  Facility  of  Errors. 

31.  If  observations  made  upon  the  same  magnitude  could 
be  repeated  under  the  same  circumstances  indefinitely,  only  a 
limited  number  of  observed  values,  which  are  exact  multiples 
of  the  least  count  of  the  instrument,  would  occur,  and  the  rela- 
tive frequency  with  which  they  occurred  would  indicate  the  law 
of  the  probability  of  the  observed  values,  that  is  to  say,  the 
law  of  facility  with  which  the  corresponding  errors  are  com- 
mitted.    In  the  theory  of  errors,  however,  it  is  necessary  to 
regard  all  observed  values  between  certain  limits  as  possible, 
so  that  when  they  are  laid  down  upon  a  line  as  abscissas,  the 
law  of  facility  may  be  represented  by  a  continuous  curve,  as 
explained  in  Art.  21.     This  is  in  fact  equivalent  to  supposing 
the  least  count  diminished  without  limit. 

The  curve  thus  obtained  is  the  probability  curve  for  an 
observed  value ;  and,  if  the  point  representing  the  true  value 
be  taken  as  origin,  the  abscissas  become  errors,  and  the  curve 
becomes  the  probability  curve  for  accidental  errors  committed 
under  the  given  circumstances. 

32.  The  probability  curves  corresponding  to  different  circum- 
stances of  observation  would  differ  somewhat,  but  in  any  case 
would  present  the  following  general  features.    In  the  first  place, 
since  errors  in  defect  and  in  excess*  are  equally  likely  to  occur, 
the  curve  must  be  symmetrical  to  the  right  and  left  of  the  point 
which  represents  the  true  value  of  the  observed  quantity.     In 
the  next  place,  since  accidental  errors  are  made  up  of  elemental 
errors  (Art.  3)  which,  as  they  may  have  either  direction,  tend 

*  There  is  usually  no  distinction  in  kind  between  these  :  either  direc- 
tion may  be  taken  as  positive,  and  errors  of  a  given  magnitude  in  one 
direction  or  the  other  are  equally  likely  to  occur. 


22 


PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  32 


to  cancel  one  another,  small  errors  are  more  frequent  than 
large  ones,  so  that  the  maximum  ordinate  occurs  at  the  central 
point.  In  the  third  place,  since  large  errors  (which  can  only 
result  when  most  of  the  elemental  errors  have  the  same  direc- 
tion and  their  greatest  magnitudes)  are  rare,  and  errors  beyond 
some  undefined  limit  do  not  occur,  the  curve  must  rapidly 
approach  the  axis  of  x  both  to  the  right  and  left,  so  that  the 
ordinate  (which  can  never  become  negative)  practically  van- 
ishes at  an  indefinite  distance  from  the  central  point. 

33.  If  y  =  <p(x)  is  the  equation  of  the  curve  referred  to  the 
central  point  as  origin,  the  general  features  mentioned  above  are 
equivalent  to  the  statements :  first,  that  y(x)  is  an  even  func- 
tion, that  is,  a  function  of  x^\  secondly,  that  9^(0)  is  its  maxi- 
mum value ;  thirdly,  that  it  is  a  decreasing  function  of  x1,  and 
practically  vanishes  when  x  is  large.  Since  it  is  impracticable 
to  select  the  function  <p  in  such  a  manner  that  <p(x~)  shall  be 
constantly  equal  to  zero  when  x  exceeds  a  certain  limit,  the  last 
condition  requires  that  the  curve  shall  have  the  axis  of  x  for  an 
asymptote ;  in  other  words,  we  must  have  </>(±  oo  )  =  o. 

When  regarded  as  the  curve  of  probability  of  an  observed 
value,  the  equation  is  y  =  <f>(x  —  a),  where  a  is  the  true  value 
of  the  observed  quantity,  the  origin  now  corresponding  to  the 
zero  point  of  the  measurements. 


The  general  form  of  the  curve  of  probability  of  an  observed 
value  will  therefore  be  similar  to  that  given  in  Fig.  3,  in  which 
A  is  the  point  whose  abscissa  a  is  the  true  value. 


§IV.]  ERROR  BETWEEN  GIVEN  LIMITS.  2$ 

The  Probability  of  an  Error  between  given  Limits. 
34.  If  the  law  of  probability  of  error  for  a  given  observation  is 

y  =  v(x) , 

the  probability  that  the  error  of  an  observation  shall  lie  between 
a  and  /3  will,  in  accordance  with  Art.  21,  be  expressed  by 


provided  that  the  value  of  this  integral  for  the  whole  range  of 
possible  errors  is  unity.  Since  we  suppose  the  function  <p(x}  to 
fulfil  the  conditions  given  in  Art.  32,  we  may  include  all  errors 
in  the  range  of  the  integral,  because  the  probability  of  large 
errors  practically  vanishes.  We  therefore  write 

> 

<p(x)dx  =  i . 

-co 

That  is  to  say,  the  whole  area  between  the  curve  and  the  axis 
in  Fig.  3  is  assumed  to  be  unity. 

35.  If  Ax  represents  the  least  count  of  the  instrument,  the 
probability  that  an  observation  shall  be  recorded  with  the  value 
x  will  be  represented  by 


E-MAz 
?(x 
_  —   Az 


If  Ax  is  so  small  that  <p(x)  may  be  regarded  as  constant  over 
the  interval,  the  value  of  this  integral  is 


The  product  <p(x}dx,  which  is  the  element  of  probability, 
being  the  element  of  the  area  which  represents  the  probability, 
is  therefore  called  the  probability  of  an  error  between  x  and 
x  +  dx,  and  is  sometimes  written  in  the  form 


f 

Jx 


24  PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  36 

The  Probability  of  a  System  of  Observed  Values. 

36.  Let  xl ,  Jtr2 ,  .  .  .  xn  be  a  series  of  observed  values  of  a 
quantity  whose  true  value  is  at  the  observations  being  all  made 
under  the  same  circumstances.     Then 

Xi  —  a,     xz  —  a,     ...     xn  —  a 
are  the  errors  of  observation;  and, 

y  =  <p(x  —  a) (i) 

being  the  law  of  facility  of  the  errors,  the  probability  before  the 
first  observation  is  made  that  x±  shall  be  the  first  observed 
value  is  <p(x±  —  d)Ax,  where  Ax  is  the  least  count  of  the 
instrument.  In  like  manner,  the  probability  that  x^  shall  be  the 
second  observed  value  is  <f>^xz  —  a}Axy  and  so  on. 

It  follows,  in  accordance  with  the  principle  explained  in  Art. 
15,  that,  if  P  denote  the  probability  of  the  compound  event  con- 
sisting in  the  occurrence  of  the  n  observed  values,  then,  before 
the  observations  were  made  we  should  have 

P  —  <p(x\  —  a)  <?(x*  —  a)  .  .  .  <p(xn  —  d)  Axn.      .     (2) 

The  Most  Probable  Value  derivable  from  a  given  System 
of  Observed  Values. 

37.  Supposing  the  form  of  the  function  <p  to  be  known,  the 
value  of  P  given  above  is  a  known  function  of  the  unknown  true 
value  a.     Regarding  different  values  of  a  as  hypotheses   all 
equally  probable  before  the  observations  were  made,  the  prin- 
ciple  enunciated  in  Art.  30  shows  that  that  value  of  a  is  most 
probable  which  assigns  to  P  the  greatest  value. 

The  value  of  a  thus  found,  or  most  probable  value,  depends 
therefore  in  part  upon  the  form  of  the  function  </>,  this  being 
the  mathematical  expression  of  a  law  which,  as  stated  in  Art.  13, 
can  never  be  absolutely  known.  We  proceed  to  the  method  of 
Gauss,  which  consists  in  determining  the  form  of  <f>  in  accord- 
ance with  which  the  arithmetical  mean  becomes  the  most  prob- 
able value. 


§IV.]  DETERMINATION  OF  THE  FORM  OF  <p.  2$ 

The  Form  of  <p  corresponding  to  the  Arithmetical  Mean. 

38.  If  we  put 

log  <p(x  —  a}  =  <l>(x  —  a),       .     .     .     .     (i) 

we  have  from  equation  (2),  Art.  36, 
logP—  <^.*\— a)  +  <p(xz— a)+  ...  +  ^On— «)  +  wlog  J^tr,   (2) 

and  #  is  to  be  so  taken  that  P,  and  therefore  log  /*,  shall  be  a 
maximum.  Hence,  putting  ^  for  the  derivative  of  4',  we  have 
by  differentiation  with  respect  to  a, 

tf(Xi  -  a)  +  V(x*  -*)+...+  $(x»  -  a)  =  o.       (3) 
Denoting  the  quantities 

xl  —  a,     Xi  —  a,     ...     xn  —  a, 

which  are  the  residuals,  by  vl ,  z/3 ,  .  .  .  vn ,  this  equation  may  be 
written 

^'OO  +  ^(fi)  +  .  -  -  +  0'On)=  o.     ...     (4) 

Supposing  now  the  value  of  a  which  satisfies  equation  (3)  to 
be  the  arithmetical  mean,  we  have  by  Art.  9, 

Vj.  +  V2  +    .  .  .    +  Vn  =  O (5) 

We  wish  therefore  to  find  the  form  of  the  function  0'  such  that 
equation  (4)  is  satisfied  by  every  set  of  values  of  vl ,  vz ,  .  .  .  vn 
which  satisfy  equation  (5).  For  this  purpose,  suppose  all  the 
values  of  v  except  v^  and  vz  to  remain  unchanged  while  equa- 
tion (5)  is  still  satisfied.  The  new  values  may  then  be  denoted 
by  z>!  -f  k  and  z>2  —  ^,  in  which  k  is  arbitrary.  Substituting  the 
new  values  in  equation  (4),  the  sum  of  the  first  two  terms  must 
remain  unchanged  since  all  of  the  other  terms  are  unchanged ; 
therefore, 

v'''fa  +  £)  +  V'''(*>2  -  K)  =  0'(zO  +  0>3); 
whence 

-    ''  '        -  4>'(v*  -  K)  ^ 


26  PROBABILITY' OF  ACCIDENTAL  ERRORS.      [Art.  38 

When  k  is  diminished  without  limit  this  becomes 


hence,  because  z^  and  z/2  are  independent,  we  infer  that 

-     -     -    .....    (7) 

where  c  is  an  unknown  constant. 

The  integral  of  equation  (7)  is  tf(v)  =  cv  -f  */:  but,  substi- 
tuting in  equation  (4),  we  find  cf  =  o;  hence 

V(v)=cv  ........     (8) 

Integrating  again, 

<!>&)  =  W  +  c", 

or,  by  equation  (i), 

log  K«0  =  -  *V  +  *",     .....     (9) 


in  which  we  have  written  —  k2  for  the  constant  \c,  because  we 
know  from  Art.  33  that  <p(v)  is  a  decreasing  function  of  z>2. 
Finally,  equation  (9)  gives 


(10) 


which  is  accordingly  the  law  of  facility  of  error  which  makes  the 
arithmetical  mean  the  most  probable  value. 

The  Determination  of  the  Value  of  C. 

39.  The  constants  C  and  h  which  arise  in  the  above  process 
are  not  independent  ;  for,  x  denoting  the  error  as  in  Art.  34,  we 
must  have 

y(x)dx  =  i  . 


Substituting  from  equation  (10)  above,  this  gives 

c(V"°xV.*r=I (I) 

J— 00 

by  which  the  value  of  C  in  terms  of  h  may  be  found. 


§IV.]          DETERMINATION  OF  THE   VALUE  OF  C.  2/ 

A  convenient  mode  of  evaluating  the  definite  integral  involved 
in  this  equation  results  from  the  consideration  of  the  solid  in- 
cluded between  the  plane  of  xy  and  the  surface  generated  by 
the  revolution  of  the  curve 


about  the  axis  of  z.     Using  polar  coordinates  in  the  plane  of 
xy  ,  the  equation  of  the  surface  is 


The  volume  of  the  solid  in  question  is  therefore  expressed  by 
either  of  the  two  formulae 

.....    (3) 

-oo-co 

and 


V=         e-rdrde  ......     (4) 

Jo  Jo 

The  second  expression  is  readily  evaluated  and  gives 


In  equation  (3),  the  limits  of  integration  are  independent  ;  hence 


Comparing  equations  (5)  and  (6),  we  have 

.....  (7) 


Substituting  in  equation  (i),we  have  C=  —T~,  and  the  law  of 
facility  becomes 


—Khfl 


*It  is  readily  shown  that     e~i*dt  —  T(J),  the  value  of  which  is  tf  IT  : 
J—  « 

equation  (7)  may  also  be  derived  by  putting  t  =  hx  in  this  result. 


28  PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  39 

a  law  which,  it  is  readily  seen,  fulfils  the  conditions  given  in 
Art.  32. 

40.  The  law  of  facility  expressed  in  the  equation  derived 
above  is  that  which  is  universally  adopted  ;  in  other  words,  it  is 
assumed  that  under  any  circumstances  of  observation  the  law  of 
facility  will  be  satisfactorily  represented  by  equation  (8)  if  the 
value  of  h  be  properly  determined.    The  mode  of  determining 
the  most  probable  value  of  h  for  a  given  set  of  observations  will 
be  given  in  the  following  section. 

We  proceed  to  develop  the  consequences  of  this  law.  Among 
them  will,  of  course,  be  found  the  rule  of  the  Arithmetical  Mean 
in  accordance  with  which  the  law  has  been  derived  (see  Art.  42). 
Certain  confirmations  of  the  law,  both  of  a  theoretic  and  a  prac- 
tical nature,  will  also  be  noticed  as  they  present  themselves. 

The  Principle  of  Least  Squares. 

» 

41.  Substituting  the  expression  now  obtained  for  the  function 
<p,  the  expression  for  the  probability  of  the  occurrence  of  the 
actual  observed  values  (as  estimated  before  the  observations 
were  made,  see  Art.  36)  becomes 


p=          e  --a-  .  .  •        «»-«      j^n  ^ 

jtt* 

This  expression,  regarded  as  a  function  of  a,  is  obviously  a 
maximum  when 

(.#!  —  of  +  (x-t  —  a)*  +  .  .  .  +  (xn  —  a?  =  a  minimum.   (2) 

Hence  the  most  probable  value  of  the  observed  quantity  a,  in 
the  case  of  observations  supposed  equally  good,  is  that  which 
assigns  the  least  possible  value  to  the  sum  of  the  squares  of  the 
residual  errors.  This  is  the  statement  in  its  simplest  form  of 
the  principle  of  Least  Squares. 

42.  The  rule  of  the  Arithmetical  Mean  follows  directly  from 
the  principle  of  Least  Squares.  Thus,  by  differentiation  with 
respect  to  a,  we  derive  from  equation  (2) 

Xi~-  a  +  Xz—  a  +  ...  -\-Xn--  <z  =  o  ; 


§IV.]  THE  PRINCIPLE  OF  LEAST  SQUARES.  2Q 

that  iSj  the  algebraic  sum  of  the  residuals  is  zero,  or 

Zx 
a  =  ^T> 

in  other  words,  the  arithmetical  mean  is  to  be  taken  as  the  most 
probable  value. 

43.  Conversely,  we  may  show  directly  that  the  arithmetical 
mean  makes  the  sum  of  the  squares  of  the  residuals  a  minimum. 
For,  if  a  is  the  arithmetical  mean,  the  residuals  are 

vl  =  Xi  —  a  ,     2/2  =  xz  —  a  ,     ...     vn  =  xn  —  a  , 

and  Zv  =  o.  Now  if  d  is  the  error  of  the  arithmetical  mean,  the 
true  value  of  the  observed  quantity  is  a  —  3,  and  the  true  ex- 
pressions for  the  errors  of  the  observed  values  are 

x:—  a  +  d  =  V-L  +  d  t     ...     xn  —  a  +  S  =  vn  +  d. 
The  sum  of«the  squares  of  the  n  errors  is  therefore 

Z(y  +  3?  =  Zv*  +  2dZv  +  nP 
=  Zv*  +  nd\ 

since  Zv  =  o.  The  minimum  value  of  this  expression  is  obvi- 
ously 2v2,  the  value  assumed  when  d  =  o  ;  that  is  to  say,  the 
sum  of  the  squares  of  the  residuals  is  least  when  the  arithmetical 
mean  is  taken  as  the  value  of  the  observed  quantity. 

The  Probability  Integral. 

44.  Taking  now  the  probability  curve  to  be 


the  probability  of  an  error  between  a  and  /?  in  magnitude  is 


and,  in  particular,  the  probability  of  an  error  numerically  less 
than  d  is 


PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  44 


If  we  put  hx  =  t,  this  may  be  written  in  the  form 


-  t 


which  shows  that  P  depends  solely  upon  the  value  of  kdy  that  is, 
upon  the  limiting  value  of  /. 

Table  I  gives  the  values  of  this  integral  for  values  of  /  from  o 
to  2  at  intervals  of  .01.     The  halves  of  the  tabular  numbers  are 
the  values  of  the  probability  of  an  error  whose  reduced  value 
falls  between  the  limits  o  and  /,  and  by  combining  these  we  can      U 
readily  find  the  values  of  the  probability  for  any  given  limits. 

The  Measure  of  Precision. 

45.  The  value  of  h  in  the  probability  curve  depends  upon  the 
circumstances  of  observation.  Let  hi  and  h^  be  the  values  of  h 
corresponding  to  two  sets  of  observations  for  which  the  curves 


FIG.  4. 

are  drawn  in  Fig.  4.     The  ordinates  corresponding  to  x  =  o 
in  the  two  curves  are  proportional  to  the  values  of  h.     Hence, 


§IV.]  THE  MEASURE  OF  PRECISION.  31 

because  small  errors  are  relatively  more  frequent  in  the  better 
set  of  observations,  the  value  of  h  for  this  set  will  be  the  larger. 
46.  Let  £j.  be  any  error,  and  put 


then,  because  dl  in  the  first  set  of  observations  and  52  in  the 
second  set  correspond  to  the  same  value  of  /  in  the  prob- 
ability integral,  equation  (3),  Art.  44,  the  probability  that  an 
error  shall  be  less  than  dl  in  the  first  set  is  the  same  as  the  prob- 
ability that  an  error  shall  be  less  than  £2  in  the  second  set.  In 
Fig.  4,  for  example,  we  have  taken  h*  =  2^  ;  it  follows  that 
dz  =  Jflj  ;  that  is  to  say,  the  probability  that  an  error  shall  not 
exceed  a  given  limit  in  the  first  case  is  the  same  as  the  prob- 
ability that  an  error  shall  not  exceed  one  half  of  the  given  limit 
in  the  second  case.*  The  ordinates  corresponding  to  dl  and  £2 
in  the  two  curves  are  drawn  in  Fig.  4.  The  areas  cut  off  in  the 
two  cases  are  equal.  It  is,  in  fact,  readily  seen  that  the  second 
curve  might  have  been  derived  from  the  first  by  reducing  the 
abscissa  of  each  point  of  the  curve  to  one  half  its  value  and  at 
the  same  time  doubling  the  corresponding  ordinate,  a  process 
which  evidently  would  not  affect  the  total  area,  which,  as  we 
have  seen,  must  always  be  equal  to  unity. 

47.  The  ratio  of  #i  and  <?2  which  correspond  to  the  same 
probability  may  be  said  to  measure  the  relative  risk  of  error  in 
the  two  cases.  Thus,  in  the  example  illustrated  in  Fig.  4,  the 
risk  of  error  in  the  first  case  is  double  that  in  the  second  case. 
It  is  natural  to  regard  the  precision  of  the  observations  in  the 
second  case  as  double  that  of  the  observations  in  the  first  case. 
So  also,  in  general,  the  ratio  of  precision  is  inversely  that  of  the 
risk  of  error  ;  that  is  to  say,  it  is  the  direct  ratio  of  the  values 
of  h,  which  are  inversely  proportional  to  the  corresponding 
values  of  3.  Accordingly  h  is  taken  as  the  measure  of  precision. 

*This  is  frequently  inaccurately  expressed  by  the  statement  that  the 
probability  of  a  given  error  in  the  first  case  is  the  same  as  that  of  the 
half  error  in  the  second  case. 


32  PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  47 

If  the  errors  in  any  system  of  observations  are  multiplied  by 
the  proper  values  of  h,  the  results  are  the  corresponding  values 
of  /.  Errors  belonging  to  different  systems  may  thus  be 
reduced  to  the  same  scale,  and  the  values  of/,  or  reduced  errors, 
will  then  admit  of  direct  comparison. 

The  Probable  Error. 

48.  The  error  which  is  just  as  likely  to  be  exceeded  as  not 
is  called  the  probable  error*     In  other  words,  the  probable 
error  is  the  value  of  d  for  which  /*=  £  in  equation  (2),  Art.  44. 
Denoting  by  p  the  corresponding  value  of  t  in  equation  (3) 
of  the  same  article,  we  have 

i          2    f  _«.  , 

—  =  —;—     e      at. 

V71  Jo 

The  solution  of  this  equation  has  been  found  to  be 
P  =  0.476936, 

which  is  the  value  of  /  corresponding  to  the  interpolated  value 
P=  0.5  in  Table  I. 

Denoting  the  probable  error  by  r,  we  have  then 

rh  =  p, 

_  p    _  0.4769 
~~h~~        h 

The  Mean  Absolute  Error. 

49.  The  mean  value  of  all  possible  errors,  having  regard  to 
their  probability  or  frequency  in  the  long  run,  is,  in  accordance 
with  Art.  24, 


*The  "probable  error"  is  thus  not  the  most  probable  error,  which  is, 
of  course,  the  error  zero,  for  which  the  ordinate  of  the  probability  curve 
is  a  maximum. 


§IV.]  THE  MEAN  ABSOLUTE  ERROR.  33 

The  value  of  this  is  of  course  zero,  the  parts  of  the  integral 
corresponding  to  positive  and  negative  errors  being  equal  and 
having  contrary  signs.  The  value  obtained  by  taking  both 
parts  of  the  integral  as  positive  is  the  mean  of  the  errors  taken 
all  positively,  or  the  mean  of  the  absolute  values  of  the  errors. 
Denoting  this  mean  by  y,  we  have 

2> 

*=v 

whence 


The  Mean  Error. 

50.  The  mean  of  all  values  of  the  square  of  the  error,  having 
regard  to  their  probabilities,  is,  in  like  manner  (see  Art.  25), 


The  error  whose  square  has  this  mean  value  is  denoted  by  e. 
On  account  of  its  importance  in  the  theory,  this  error  is  called 
the  mean  error.  Thus 


e>=     * 


The  value  of  the  definite  integral  involved  in  this  expression 
may  be  deduced  from  the  result  found  in  Art.  39,  equation  (7), 
namely, 


Differentiating  with  respect  to  h,  we  have 

hf^-v*dx_        ^ 

and,  substituting  in  the  value  of  e2,  we  find 


i  i 

—  — Tl>       °r       =   ~  ~i — / 

* 


34  PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  51 

Measures  of  the  Risk  of  Error. 

51.  We  have  seen  in  Art.  47  that  the  errors  corresponding  in 
two  different  systems  to  the  same  value  of  the  reduced  error  / 
measure  by  their  ratio  the  comparative  risk  of  error  in  the 
two  systems.  Thus  the  error  corresponding  to  any  fixed  value 
of  /  might  be  taken  as  the  measure  of  this  risk.  Accordingly 
either  of  the  errors 


which  correspond  respectively  to  the  reduced  errors 

i  i 

pt      V*'      V^' 

may  be  taken  as  the  measure  of  the  risk  of  error*  or  inverse 

measure  of  precision. 

The  probable  error  r  is  that  which  is  most  frequently  em- 
ployed in  practice.  Each  of  the  others  bears  a  fixed  ratio  to  r, 
their  values  being  respectively 

*The  error  #  was  called  by  De  Morgan  the  mean  risk  of  error,  because 
it  is  the  mean  expectation  of  error,  using  the  term  expectation  in  the 
same  sense  as  in  the  expression  "value  of  an  expectation."  (See  foot- 
note on  page  15.)  It  corresponds  to  what  is  generally  called  in  annuity 
tables  "the  expectation  of  life  "  for  persons  of  a  given  age,  which  should 
rather  be  called  "the  mean  duration  of  survival"  for  persons  of  the 
given  age.  On  the  other  hand,  the  probable  error  r  is  analogous  to  the 
remaining  term  for  which  a  person  of  the  given  age  is  as  likely  as  not  to 
live.  This  might  be  called  "the  probable  term  of  survival,"  and  its 
value  may  differ  materially  from  the  mean  duration.  Thus,  according  to 
the  Carlisle  mortality  table,  one  half  of  the  whole  number  of  persons 
thirty  years  old  survive  for  the  term  of  36.6  years,  but  the  mean  duration 
of  life  for  such  persons,  as  computed  from  the  same  table,  is  only  34.3 
years.  This  indicates  that  the  law  of  mortality  is  such  that  the  half 
which  exceed  the  term  of  probable  survival  do  so  by  a  total  amount  less 
than  that  by  which  the  other  half  fall  short  of  it. 

In  the  case  of  errors  the  difference  falls  in  the  opposite  direction.  In 
the  long  run  one  half  of  the  errors  exceed  r  ;  and  the  fact  that  ij  >  r 
shows  that  the  half  which  exceed  r  do  so  by  a  total  amount  greater  than 
that  by  which  the  other  half  fall  short  of  it. 


§IV.] 


MEASURES  OF  THE  RISK  OF  ERROR. 


35 


=  1.1829  r, 


(i) 


=  14826  r, (2) 


52.  Fig.  5  shows  the  positions  of  the  ordinates  corresponding 
to  r,  TJ  and  s  in  the  curve  of  facility  of  errors 


The  diagram  is  constructed  for  the  value  h  =  2. 


FIG.  5. 


From  the  definitions  of  the  errors  it  is  evident  that  the  ordinate 
of  r  bisects  the  area  between  the  curve  and  the  axes,  that  of  TJ 
passes  through  its  centre  of  gravity,  and  that  of  e  passes  through 
its  centre  of  gyration  about  the  axis  of  y. 


36  PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  52 

The  advantage  of  employing  in  practice  a  measure  of  the 
risk  of  error,  instead  of  the  direct  measure  of  precision,  results 
from  the  fact  that  it  is  of  the  same  nature  and  expressed  in  the 
same  units  as  the  observations  themselves.  It  therefore  conveys 
a  better  idea  of  the  degree  of  accuracy  than  is  given  by  the  value 
of  the  abstract  quantity  h.  When  the  latter  is  given,  it  is  of 
course  necessary  also  to  know  the  unit  used  in  expressing  the 
errors. 

Tables  of  the  Probability  Integral. 

53.  The  integral     e~^dt  is  known  as  the  error  function  and 

Jo 

is  denoted  by  Erf/.*  Table  I,  which  has  already  been  described, 
Art.  44,  gives  the  values  °f-j—  Erf  /,  which  is  the  probability 

that  an  error  shall  be  numerically  less  than  the  error  x,  of 
which  the  reduced  value  is  /.  The  argument  of  this  table  is 
the  reduced  error  t. 

But  it  is  convenient  to  have  the  values  of  the  probability 
given  also  for  values  of  the  ratio  of  the  error  x  to  the  probable 
error.  Putting  z  for  this  ratio,  we  have,  since  hx  =  /  and  hr=p, 

2=x_  =  ± 
r  ~    p  ' 

Table  II  gives,  to  the  argument  z,  the  same  function  of  /  which 
is  given  in  Table  I  ;  that  is  to  say,  the  function  of  z  tabulated  is 


*The  integral     e     *dt  is  denoted  by  Erfc  /,  being  the  complement  of 
the  error  function,  so  that 

Erf  /  +  Erfc  /  =  \e~^  dt  =  $  V  T. 
Jo 

These  functions  occur  in  several  branches  of  Applied  Mathematics. 
A  table  of  values  of  Erfc  /  to  eight  places  of  decimals  was  computed  by 
Kramp  ("  Analyse  des  Refractions  Astronomiques  et  Terrestres,"  Stras- 
bourg, 1799),  an(l  from  tnis  the  existing  tables  of  the  Probability  Integral 
have  been  derived. 


§IV.]       TABLES  OF  THE  PROBABILITY  INTEGRAL.  $? 

which  is  the  probability  that  an  error  shall  be  numerically  less 
than  the  error  x  whose  ratio  to  the  probable  error  is  z. 

54.  By  means  of  the  tables  of  the  probability  integral,  com- 
parisons have  been  made  between  the  actual  frequency  with 
which  given  errors  occur  in  a  system  containing  a  large  number 
of  observations  and  their  probabilities  in  accordance  with  the 
law  of  facility. 

The  following  example  is  given  by  Bessel  in  the  Fundamenta 
Astronomiae.  From  470  observations  made  by  Bradley  on  the 
right  ascensions  of  Procyon  and  Altair,  the  probable  error  of 
a  single  observation  was  found  (by  the  formula  given  in  the  next 
section)  to  be 


With  this  value  of  r,  the  probability  that  an  error  shall  be 
numerically  less  than  o".i  is  found  by  entering  Table  II  with 
the  argument 


and  the  probability  that  it  shall  be  less  than  o".2,  o"-3  and  so 
on,  by  entering  the  table  with  the  successive  multiples  of  this 
quantity.  In  the  annexed  table  the  first  column  contains  the 
successive  values  of  the  limiting  error  x,  the  second  those  of  zt 

Theoretical        Actual 


X 

2 

p 

Differences. 

IN  OS.  Ot 

Errors. 

IN  OS.  I 

Error 

o".i 

0-379 

0.2018 

0.2OI8 

94.8 

94 

0   .2 

0.758 

0.3907 

0.1889 

88.8 

88 

o  -3 

I.I38 

0.5573 

0.1666 

78-3 

78 

o  .4 

I-5I7 

0.6937 

0.1364 

64.1 

58 

o  -5 

1.896 

0.7990 

0.1053 

49-5 

5i 

o  .6 

2.275 

0.8751 

0.0761 

35-8 

36 

o  .7 

2.654 

0.9265 

0.0514 

24.2 

26 

o  .8 

3-034 

0.9593 

0.0328 

15-4 

H 

o  .9 

3-4I3 

0.9787 

0.0194 

9.1 

10 

I    .0 

3-792 

0.9894 

0.0107 

5-0 

7 

00 

00 

I.OOOO 

0.0106 

5-o 

8 

38  PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  54 

and  the  third  the  corresponding  values  of  the  probability  of  an 
error  less  than  x  as  given  by  Table  II.  The  fourth  column 
contains  the  successive  differences  of  these,  so  that  each  of  the 
numbers  contained  in  it  is  the  probability  of  an  error  falling 
between  the  corresponding  value  of  x  and  that  which  precedes 
it.  The  fifth  column  contains  the  multiples  of  these  by  470, 
which  are  the  theoretical  numbers  of  errors  to  be  expected 
within  the  intervals,  the  last  number  in  the  column  being  the 
number  of  errors  which  should  exceed  i".o.  Finally,  the  last 
column  contains  the  actual  numbers  of  errors  which  occurred  in 
the  corresponding  intervals,  as  given  by  Bessel.  The  agree- 
ment between  the  theoretical  and  actual  numbers  is  remarkably 
close,  and  forms  a  practical  confirmation  of  the  adopted  law  of 
facility. 

The  Distribution  of  Errors  on  a  Plane  Area. 

55.  The  deviations  of  the  bullet  marks  in  target  practice  from 
the  point  aimed  at  are  of  the  nature  of  accidental  errors.  It  is 
usually  assumed  that  the  lateral  deviations  and  the  vertical 
deviations  are  independent  of  one  another,  and  that  each  follows 
the  law  of  facility  for  linear  errors.  We  proceed  to  determine 
the  resulting  law  of  the  distribution  of  the  shots  upon  the  plane 
area. 

Let  the  point  aimed  at  be  taken  as  the  origin  of  coordinates, 
the  horizontal  deviation  of  a  shot  being  denoted  by  x  and  the 
vertical  deviations  byjy,  and  let  these  deviations  be  assumed  to 
have  the  same  measure  of  precision.  Then  the  probability  of 
a  horizontal  deviation  between  x  and  x  +  dx  is 


and  for  each  value  of  x  the  probability  of  a  vertical  deviation 
between^/  andjy  +  dy  is 


§  IV.]   DISTRIBUTION  OF  ERRORS  ON  A  PLANE  AREA.        39 

Hence  the  probability  of  hitting  the  elementary  rectangular 
area  dxdy,  which  involves  the  joint  occurrence  of  these  devia- 
tions, is 


and,  since  the  probability  of  hitting  an  elementary  area  is  pro- 
portional to  the  area,  if  «  denote  such  an  area  situated  at  the 
point  (x  ,  y),  the  probability  of  hitting  it  is 


where  r  denotes  the  distance  of  a.  from  the  origin. 

Thus  the  hypothesis  of  independent  vertical  and  horizontal 
deviations,  each  following  the  usual  law  of  facility  and  having 
the  same  measure  of  precision,  leads  to  the  conclusion  that  the 
facility  of  the  resultant  deflection  depends  solely  upon  its  linear 
amount,  rt  and  not  at  all  upon  its  direction.*  This  agrees  with 

*  Sir  John  Herschel's  proof  of  the  law  of  facility  of  errors  (Edinburgh 
Review,  July,  1850)  rests  upon  the  assumption  that  it  must  possess  the 
property  which  is  above  shown  to  belong  to  the  exponential  law.  He 
compares  accidental  errors  to  the  deviations  of  a  stone  which  is  let  fall  with 
the  intention  of  hitting  a  certain  mark,  and  assumes  that  the  deviations  in 
the  directions  of  any  two  rectangular  axes  are  independent.  But,  since 
there  is  no  reason  why  the  resultant  deviations  should  depend  upon  their 
direction,  this  implies  that,/(*2)  being  the  law  of  facility,  we  must  have 


where  x/  and^7  denote  coordinates  referred  to  a  new  set  of  rectangular 
axes,  so  that 

X*  +  y*  —  x»  +  y. 

Now  the  solution  of  the  functional  equation 

A*}/W=i<ft 

is 


where  c  and  k  are  constants. 

There  is  no  a  priori  reason  why  the  deviations  in^y  should,  as  assumed 


40  PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Ait.  55 

the  usual  custom  of  judging  of  the  accuracy  of  a  shot  solely  by 
its  distance  from  the  point  aimed  at. 

The  Surface  of  Probability. 

56.  If  at  every  point  of  the  plane  of  xy  we  erect  a  perpendicular 
z%  taking 


we  shall  have  a  surface  of  probability  analogous  to  the  curve  of 
probability  in  the  case  of  linear  errors.  Since  the  probability 
of  hitting  the  elementary  area  dxdy  is  zdxdy,  the  probability 
of  hitting  any  area  is  the  value  of  the  double  integral 


ii 


zdxdy 


taken  over  the  given  area.  That  is  to  say,  it  is  the  volume  of 
the  right  cylinder  having  this  area  for  its  base,  and  having  its 
upper  surface  in  the  surface  of  probability. 

The  probability  surface  is  a  surface  of  revolution.  The  solid 
included  between  it  and  the  plane  of  xy  is  in  fact  similar  to 

Too 

that  employed  in  Art.  39,  in  evaluating  the  integral    e~™x*dx. 

J — 00 

The  Probability  of  Hitting  a  Rectangle. 

57.  The  probability  of  hitting  the  rectangle  included  between 
the  horizontal  lines y  —y,,y—yz  and  the  vertical  lines  x  =  xlt 
x  =  x*  is  the  double  integral 


*  r  !*,- 

*    -k  J3/i 


above,  occur  with  the  same  relative  frequency  when  x  has  one  value  as 
when  it  has  another  ;  but  it  is  noteworthy  that,  having  made  this  assump- 
tion, no  other  law  of  facility  of  linear  deviation  would  produce  a  law  of 
distribution  in  area  involving  only  the  distance  from  the  centre.  On  the 
other  hand,  no  other  law  of  distribution  in  area  depending  only  upon  r 
(such  for  example  as  e~r)  would  make  the  law  of  facility  for  deviations 
iny  independent  of  the  value  of  x. 


§IV.]      PROBABILITY  OF  HITTING  A  RECTANGLE.  4! 

which,  because  the  limits  for  each  variable  are  independent  ot 
the  other,  is  equivalent  to 


-*.«. 


that  is,  it  is  the  product  of  the  probabilities  that  x  and  y  respec- 
tively shall  fall  between  their  given  limits.  This  result  is,  of 
course,  nothing  more  than  the  expression  of  the  hypothesis 
made  in  Art.  55.*  If  h  be  known,  the  values  of  the  factors  in 
the  expression  (2)  may  be  derived  from  Table  I,  as  explained 
in  Art.  44. 

In  particular,  putting  xv  =  —  3,  x.2  =  8,  y^  =  —  df,  yz  =  df, 
we  have  for  the  probability  of  hitting  a  rectangle  whose  centre 
is  at  the  origin  and  whose  sides  are  2d  and  28'  , 

p  =  />s/V, 

where  PS  and  /V  are  tabular  results  taken  from  Table  I,  if  h  be 
given,  or  from  Table  II  if  the  probable  error  of  the  deviations 
be  given. 

For  example,  for  the  square  whose  centre  is  the  origin  and 
whose  half  side  is  rt  ,  the  probable  error  of  the  component 
deviations,  the  probability  of  hitting  is  J. 

Again,  to  find  the  side  of  the  centrally  situated  square  which 
is  as  likely  as  not  to  be  hit,  and  which  therefore  may  be  called 
the  probable  square,  we  must  determine  the  value  of  d  for 
which  Ps  =  V  i  =  0.7071.  This  will  be  found  to  correspond  to 
/=  0.7437,  whence  the  side  of  the  square  is  2<5,  where 


*The  property  of  the  probability  surface  corresponding  to  the  assump- 
tion that  the  relative  frequency  of  the  deviations  in  y  is  independent  of 
the  value  of  x  is  that  any  section  parallel  to  the  plane  of  yz  may  be 
derived  from  the  central  section  in  that  plane  by  reducing  all  the  values 
of  0  in  the  same  ratio.  In  accordance  with  the  preceding  foot-note,  this 
is  the  only  surface  of  revolution  possessing  this  property. 


42  PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  58 

The  Probability  of  Hitting  a  Circle. 

58.  Putttng  a  =  2xrdr  in  the  expression  derived  in  Art.  55, 
the  probability  of  hitting  the  elementary  annular  area  between 
the  circumferences  whose  radii  are  r  and  r  +  dr  is  found  to  be 


dp  =  z&e-rdr  .......    (i) 

Hence  the  probability  that  the  distance  of  a  shot  from  the  point 
aimed  at  shall  fall  between  r^  and  rz  is 

e~^  -  g-"*.  (2) 

Putting  the  lower  limit  r±  equal  to  zero,  we  have,  for  the  prob- 
ability of  planting  a  shot  within  the  circle  whose  radius  is  ry 


(3) 


a  formula  in  which  h  is  the  measure  of  the  accuracy  of  the 
marksman. 

The  Radius  of  the  Probable  Circle. 

59,  If  we  denote  by  a  the  value  of  r  corresponding  to/  =  \ 
in  equation  (3)  of  the  preceding  article,  we  shall  have 


whence 


Then  a  is  the  radius  of  the  probable  circle,  that  is,  the  circle 
within  which  a  shot  is  as  likely  as  not  to  fall,  or  within  which 
in  the  long  run  the  marksman  can  plant  half  his  shots.  Thus 
a  is  analogous  to  the  probable  error  in  the  case  of  linear  devia- 
tions, and,  being  inversely  proportional  to  ht  may  be  taken  as 
an  inverse  measure  of  the  skill  of  the  marksman. 

Eliminating  h  from  the  formula  for/  by  means  of  equation  (i), 
we  obtain 


§  IV.]        THE  RADIUS  OF  THE  PROBABLE  CIRCLE.  43 

Denoting  by  n  the  whole  number  of  shots,  and  by  m  the 
number  of  those  which  miss  a  circular  target  of  radius  ry  we 
may,  if  n  and  m  be  sufficiently  large,  put 

m 

Supposing/  in  equation  (3)  to  be  thus  determined,  we  derive 
the  formula 

log  2 


log  n  —  log  m  ' 
in  which  the  ordinary  tabular  logarithms  may  be  employed.* 

The  Most  Probable  Distance. 

60.  Equation  (i),  Art.  58,  shows  that  the  probability  of  hitting 
the  elementary  annulus  of  radius  r  is  proportional  to 

«-**. 

The  value  of  r  which  makes  this  function  a  maximum  is  found 
to  be  identical  with  e,  the  mean  error  of  the  linear  deviations, 
namely, 


which  is  therefore  the  most  probable  distancef  at  which  a  shot 
can  fall. 

This  distance  might,  like  a,  be  taken  as  the  inverse  measure 
of  the  skill  of  the  marksman. 

*This  is  Sir  John  Herschel's  formula  for  the  inverse  measure  of  the 
skill  of  the  marksman.  See  "  Familiar  Lectures  on  Scientific  Subjects," 
p.  498.  London  and  New  York,  1867. 

fThe  point  at  which  the  probability  is  a  maximum  (that  is,  where  the 
density  of  the  shots  in  the  long  run  is  the  greatest)  is  of  course  the 
origin,  at  which  the  ordinate  z  in  the  probability  surface  is  a  maximum. 
The  value  of  r  here  determined  is  that  for  which  the  right  cylindrical 
surface  included  between  the  plane  of  xy  and  the  probability  surface  is  a 
maximum,  that  is,  the  annulus  which  contains  the  greatest  number  of 
shot  in  the  long  run, 


44  PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  61 

Measures  of  the  Accuracy  of  Shooting. 

6l.  Any  quantity  inversely  proportional  to  h  might  be  taken 
as  the  measure  of  the  marksman's  risk  of  error,  or  inverse 
measure  of  precision.  We  may  employ  for  this  purpose  either 
a,  the  radius  of  the  probable  error,  e,  the  most  probable  dis- 
tance, £,  the  half  side  of  the  probable  square  (Art.  57),  or  rlt  the 
probable  error  of  a  linear  deviation. 

The  most  probable  value  of  h  derivable  from  n  given  shots 
will  be  shown  in  the  next  section,  Art.  73,  to  be 


Employing  this  value  of  h  we  have 

«  =  ^!p  =  0.83,6^, 

IV 


*  =  2^3Z    =  0.7437^, 
*  =  T          =0-4769^. 


Examples. 

1.  Show  that  the  abscissa  of  the  point  of  inflexion  in  the 
probability  curve  is  the  mean  error. 

2.  In  1000  observations  of  the  same  quantity  how  many  may 
be  expected  to  differ  from  the  mean  value  by  less  than   the 
probable  error,  by  less  than  the  mean  absolute  error,  and  by 
less  than  the  mean  error  respectively?  500,  575,  683. 

3.  An  astronomer  measures  an  angle  100  times  ;  if,  when  the 
unit  employed  is  i",  the  measure  of  precision  is  known  to  be 


§  IV.]  EXAMPLES.  45 

h  =  -|-,  how  many  errors  may  be  expected  to  have  a  numerical 
value  between  2"  and  4"  ?  31. 

4.  In  125  observations  whose  probable  error  is  2",  how  many 
errors  less  than  i"  are  to  be  expected  ?  33. 

5.  If  the  probable  error  is  ten  times  the  least  count  of  the 
instrument,  show  that  about  27  observations  out  of  1000  will  be 
recorded  with  the  true  value,  and  21  will  exceed  it  by  an  amount 
equal  to  the  probable  error. 

6.  If  h  is  changed  to  mh  (m  >  i),  errors  less  than  a  certain 
error  ^  are  more  probable,  and  errors  greater  than  xl  are  less 
probable.     Find  4  the  reduced  value  of  x^  . 


7.  Show  that  the  envelop  of  the  probability  curve,  when  h 
varies,  is  the  hyperbola 


the  abscissa  of  the  point  of  contact  being  the  mean  error. 
8.  Show  that 


and  thence  derive  the  value  of  the  integral. 

9.  Deduce  the  formula  of  reduction  (m  positive) 


and  thence  show  that  (n  being  a  positive  integer)  the  mean 
value  of  the  2#th  power  of  the  error  is 


and  that  the  mean  absolute  value  of  the  (in  +  i)th  power  of 
the  error  is 

n\ 

t  in  4.  i     ,       • 

h     h      * 


46  PROBABILITY  OF  ACCIDENTAL  ERRORS.      [Art.  61 

10.  Show  that 


325        3     7 
ii.  Deduce  the  formula  of  reduction  (n  positive) 


and  thence  show  that 


12.  Find  the  probability  that  the  deviation  of  a  shot  shall 
exceed  20,.  ^. 

13.  Find  the  probability  that  a  shot  shall  fall  within  the  circle 
whose  radius  is  e.  i  —  e~*  =  0.3935. 

14.  A  marksman  shoots  500  times  at  a  target  ;  if  his  skill  is 
such  that  when  errors  are  measured  in  feet,  h  —  i,  what  is  the 
number  of  bullet  marks  between  two  circles  described  from  the 
centre  with  radii  i  and  2  feet?  175. 

15.  If  errors  are  measured  in  inches  in  example  14,  what  are 
the  values  of  h  and  of  a  ?  ^,  9.99. 

1  6.  An  archer  is  observed  to  plant  9  per  cent  of  his  arrows 
within  a  circle  one  foot  in  diameter  ;  what  is  the  diameter  of 
a  target  which  he  might  make  an  even  bet  to  hit  at  the  first 
shot  ?  2  ft.  8$  in. 

17.  A  hits  a  target  3  feet  in  diameter  51  times  out  of  79  shots  ; 
B  hits  one  2  feet  in  diameter  39  times  out  of  87  shots.     Find  the 
diameters  of  the  targets  that  each  can  make  an  even  wager  to 
hit  at  the  first  shot.  For  A,  2.45  feet  ;  for  B,  2.16  feet. 

18.  In  example  17,  what  are  the  odds  that  B  will  hit  A's 
probable  circle  at  the  first  shot?  About  59  to  41. 

19.  If  the  circular  target  which  a  marksman  has  an  even 
chance  of  hitting  be  divided  by  circumferences  cutting  the  radius 
into  four  equal  parts,  how  many  shots  out  of  1000  will  fall  in  the 
respective  areas  ?  42,  117,  164,  177. 


§  IV.]  EXAMPLES.  47 

20.  A  circular  target  32  inches  in  diameter  is  divided  into 
rings  by  circumferences  cutting  the  radius  into  four  equal  parts. 
The  number  of  shots  out  of  1000  which  fell  in  the  several  areas 
were  31,  89,  121,  141 ;  what  are  the  respective  values  of  a  in 
inches  determined  From  the  numbers  of  shots  in  the  several 
circles?  18.764,  18.628,  19.025,19.202. 

21.  Find  the  probability  of  hitting  a  square  target  circum- 
scribing the  circle  whose  radius  is  a.  »579O- 

22.  If  several  shots  be  fired  at  a  wafer  on  a  wall  and  the  wafer 
be  subsequently  removed,  show  that  the  centre  of  gravity  of 
the  shot  marks  is  the  most  probable  position  of  the  wafer. 


V. 

THE  COMBINATION  OF  OBSERVATIONS  AND  PROBABLE 
ACCURACY  OF  THE  RESULTS. 

The  Probability  of  the  Arithmetical  Mean. 

62.  We  have  seen  that,  in  accordance  with  the  law  of  facility 
which  we  have  adopted,  the  best  result  of  the  combination  of  a 
number  of  equally  good  observations  is  their  arithmetical  mean. 
We  have  next  to  determine  the  probable  accuracy  of  this  result, 
and  then  to  consider  the  best  method  of  combining  observations 
of  unequal  precision. 

Let  there  be  n  observations,  the  law  of  facility  of  error  for 
each  of  which  is 


a  being  the  true  value  of  the  observed  quantity,  and  x±  ,  x«  .  .  .  xn 
the  observed  values.  Then  the  value  of  P,  equation  (2),  Art. 
36,  becomes 


and,  as  shown  in  Art.  30,  the  probabilities  of  the  different 
hypotheses  which  we  can  make  as  to  the  value  of  a  are  propor- 
tional to  the  corresponding  values  of  P. 

63.  Let  us  now  take  a  to  denote  the  arithmetical  mean,  and 
put  a  —  S  for  the  true  value,  so  that  d  is  the  error  of  the  arith- 
metical mean  ;  then  denoting  the  residual  by  v,  the  true  error 
will  be  x  —  a  +  d  =  v  +  d.  It  was  shown  in  Art.  43  that 


hence  the  general  value  of  P  must  now  be  written 
p=».e-»W'ljx. 


§  V.J     PROBABILITY  OF  THE  ARITffME  TICAL  MEAN.        49 

and  the  value  expressed  by  equation  (2)  is  now  the  maximum 
value,  corresponding"  to  <5  =  o.  Distinguishing  this  value  by  the 
symbol  P0,  equation  (3)  may  be  written 


(4) 

Since  the  probability  of  <S,  which  is  the  error  of  our  final 
determination,  is  proportional  to  P,  and  P0  is  independent  of  «5, 
equation  (4)  shows  that  the  arithmetical  mean  has  a  law  of  prob- 
ability which  is  identical  with  that  which  we  have  adopted  in 
equation  (i)  for  the  single  observations,  except  that  nh2  takes 
the  place  of  h\  Thus,  denoting  by  yQ  the  facility  of  error  in  the 
arithmetical  mean,  we  have 


(5) 


The  fact  that  the  assumption  of  the  law  (i)  for  a  single 
observation  implies  a  law  of  the  same  form  for  the  final  value 
determined  from  the  combined  observations  is  one  of  the  con- 
firmations of  this  law  alluded  to  in  Art.  40.* 

64.  Equation  (5)  of  the  preceding  article  shows  that  the 
arithmetical  mean  of  n  observations  may  be  regarded  as  an 
observation  made  with  a  more  precise  instrument,  the  new 
measure  of  precision  being  found  by  multiplying  that  of  the 
single  observations  by  V  n.  Since  hr  is  constant  when  r  repre- 
sents any  one  of  the  measures  of  risk,  we  have  for  the  probable 
error  of  the  arithmetical  mean, 

r 

r°  =7^> 

*In  general,  an  assumed  law,  y  =  <j>(x),  of  facility  of  error  for  the 
single  observations  would  produce  a  law  of  a  different  form  for  the  result 
determined  from  n  observations.  Laplace  has  shown  that  whatever  be 
the  form  of  0  for  the  single  observations,  the  law  of  facility  of  error  in 
the  arithmetical  mean  approaches  indefinitely  to 

y  =  ce~™    ' 

as  a  limiting  form,  when  n  is  increased  without  limit.  See  the  memoir 
"  On  the  Law  of  Facility  of  Errors  of  Observation,  and  on  the  Method  of 
Least  Squares,"  by  J.  W.  L.  Glaisher,  Memoirs  Royal  Ast.  Soc.,  vol.  xxxix 
pp.  104,  105. 


5O  THE  COMBINATION  OF  OBSERVATIONS.       [Art.  64 


and  the  same  relation  holds  in  the  case  of  either  of  the  other 
measures  of  risk. 

Thus,  for  example,  it  is  necessary  to  take  four  observations 
in  order  to  double  the  precision,  or  reduce  the  risk  of  error  to 
one  half  its  original  value. 

The  probable  error  of  a  final  result  is  frequently  written 
after  it  with  the  sign  ±.  Thus,  if  the  final  determination  of 
an  angle  is  given  as  36°  42'.  3  ±  i'.22,  the  meaning  is  that  the 
true  value  of  the  angle  is  exactly  as  likely  to  lie  between  the 
limits  thus  assigned  (that  is,  between  36°  4i'.o8  and  36°  43'.  52) 
as  it  is  to  lie  outside  of  these  limits. 

The  Combination  of  Observations  of  Unequal  Precision. 

65.  When  the  observations  are  not  equally  good,  let  fa,  k2, 
...  hn  be  their  respective  measures  of  precision  ;  so  that,  a  being 
the  true  value,  the  facility  of  error  of  x±  is 


*  =  ., 

that  of  Xi  is 


and  so  on.  The  value  of  P,  Art.  36,  which  expresses  the  prob- 
ability of  the  given  system  of  observed  values  on  the  hypothesis 
of  a  given  value  of  #,  now  becomes 

*.  .  .  .  J*B;      .    (I) 


and,  as  before,  the  probabilities  of  different  values  of  a  are  pro- 
portional to  the  values  they  give  to  P. 

It  follows  that  that  value  of  a  is  most  probable  which  makes 
Ih\x  -  af  or 

h\(x\—a?  +  J%(Xz—  «)2+  ...  -\-h\(xn—af—  a  minimum.  (2) 

In  other  words,  if  the  error  of  each  observation  be  multiplied  by 
the  corresponding  measure  of  precision,  so  as  to  reduce  the  errors 


§V.]          OBSERVATIONS  OF  UNEQUAL  PRECISION.  gt 

to  the  same  relative  value  (see  Art.  47),  it  is  necessary  that  the 
sum  of  the  squares  of  the  reduced  errors  should  be  a  minimum. 
This  is,  in  fact,  the  more  general  statement  of  the  principle  of 
Least  Squares. 

Differentiating  with  respect  to  a,  we  have 

h\  (X  -  a)  +  hi  (*,  -  a)  +  ...  +  ft  (*»  -  «)  =  o;      (3) 
and  the  value  of  a  determined  from  this  equation  is 

_  h\Xi  +  h\xt  +  .  .  .  +  hlxn  _  Zh*x  , 

h\  +  h\  +  .  .  .  +  ft  ' 


which  is  therefore  the  most  probable  value  of  a  which  can  be 
derived  from  the  n  observations. 

Weights  and  Measures  of  Precision. 

66.  The  value  of  a  found  above  is  in  fact  the  weighted 
arithmetical  mean  of  the  observed  values  (see  Art.  11),  when  the 
respective  values  of  h2  are  taken  as  the  weights.  But,  since  the 
weights  are  numbers  with  whose  ratios  only  we  are  concerned, 
we  may  use  any  proportional  numbers  A  ,  A  ,  .  .  .  pn  ,  in  place 
of  the  values  of  h.  Thus  putting 


.    .    (5) 
equation  (4)  may  be  written 


Hence  the  most  probable  value  which  can  be  derived  from  the 
n  observations  is  the  weighted  arithmetical  mean,  the  weights 
of  the  observations  being  proportional  to  the  squares  of  their 
measures  of  precision. 

,  The  quantity  h  in  equations  (5)  is  the  measure  of  precision  of 
an  observation  whose  weight  is  unity.  It  is  immaterial  whether 
such  an  observation  actually  exists  among  the  n  observations  or 
not. 


52  THE  COMBINATION  OF  OBSERVATIONS.      [Art.  66 

If  each  of  the  observations  has  the  weight  unity,  2p  takes  the 
value  n,  and  the  value  of  a  becomes  the  ordinary  arithmetical 
mean. 

The  Probability  of  the  Weighted  Mean. 

67.  Let  us  now,  employing  a  to  denote  the  value  determined 
above,  put  a  +  d  in  place  of  a  in  the  value  of  Pt  so  that  3  repre- 
sents the  error  in  our  final  determination  of  a.  Then,  writing 
v  for  the  residual,  we  have,  as  in  Art.  63,  to  replace  x  —  a  by 
v  +  <?.  The  value  of/*,  equation  (i),  Art.  65,  thus  becomes 


p  =     .     n  e  _  »,  ,.  +  „.  J 

7TS™ 

Now,  by  equation  (3),  2h*v  =  o,  therefore 

I/i9  (v  +  <5)2  =  JAV  +  ^l^3  • 
substituting,  we  obtain 

P=  *'*•  —  *»  .-^-"^'J^,  .  .  .  J*n. 

Hence,  putting  /o  for  the  value  assumed  by  P  when  d  =  o,  we 

have 

p=poe-**iki  +  **  +  --  +  Wf 

Since  the  probability  of  $  is  proportional  to  P,  it  follows,  as 
in  Art.  63,  that  the  law  of  facility  of  the  mean  is  of  the  same 
form  as  those  of  the  separate  observations,  the  square  of  the 
new  measure  of  precision  being  the  sum  of  the  squares  of  those 
of  the  separate  observations.  Denoting  the  facility  of  error  in 
the  weighted  mean  by  J0,  and  employing  the  notation  of  Art 
66,  we  have  therefore 


in  which  h  is  the  measure  of  precision  of  an  observation  whose 


§V.]         PROBABILITY  OF  THE   WEIGHTED  MEAN.  53 

weight  is  unity.     When  the  weights  are  all  equal,  this  formula 
becomes  identical  with  that  of  Art.  63. 

68.  The  weight  of  the  mean  is  defined  in  Art.  12  to  be  J?/>, 
the  sum  of  the  weights  of  the  constituent  observations.  Hence 
the  value  of  yQ  found  above  shows  that,  in  comparing  the  final 
result  with  any  single  observation,  as  well  as  in  comparing  the 
observations  with  one  another,  the  measures  of  precision  are 
proportional  to  the  square  roots  of  the  weights. 

The  probable  error  being  inversely  proportional  to  h,  it  fol- 
lows that,  r  representing  the  probable  error  of  an  observation 
whose  weight  is  unity,  and  r0  that  of  the  mean  whose  weight  is 
2p}  we  shall  have 

r 


This  result  includes  that  of  Art.  64,  and,  like  it,  is  applicable 
to  either  of  the  measures  of  risk. 

The  Most  Probable  Value  of  h  derivable  from  a  System  of 
Observations. 

69.  Substituting  the  values  of  h±  ,  h^  ,  .  .  .  hn  in  terms  of  the 
weights,  equations  (5),  Art.  66,  the  value  of  P,  equation  (i), 
Art.  65,  becomes 

r,  .;.-4fc.     (i) 


The  same  principle  which  we  have  employed  to  determine 
the  most  probable  value  of  the  observed  quantity  serves  to 
determine  the  most  probable  value  of  h.  Thus  the  most  prob- 
able value  of  h  is  that  which  gives  the  greatest  value  to  P,  or, 
omitting  factors  independent  of  h,  to  the  expression 


x  —  o)a 

Putting  the  derivative  of  this  expression  equal  to  zero,  we 
have 


54  THE  COMBINATION  OF  OBSERVATIONS.      [Art.  69 

whence 


in  which  a  denotes  the  true  value  of  the  observed  quantity. 
70.  Equation  (2)  may  be  written 


When  the  observations  are  all  made  under  the  same  circum- 
stances, so  that  we  may  put 


the  equation  becomes 

a^-i  .......  <» 

in  which  h  denotes  the  measure  of  precision  of  each  of  the 
observations.  The  second  member  of  this  equation  is  the  value 
of  e2,  the  square  of  the  "  mean  error,"  which  was  defined  in 
Art.  50  as  the  mean  value  of  the  square  of  the  error,  having 
regard  to  its  probability  in  a  system  of  observations  whose 
measure  of  precision  is  Ji.  In  other  words,  it  is  the  mean 
squared  error  in  an  unlimited  number  of  observations  made 
under  the  given  circumstances  of  observation. 

On  the  other  hand,  the  first  member  of  equation  (2)  is  the 
actual  mean  squared  error  for  the  n  given  observations.  The 
square  root  of  this  quantity  may  be  called  the  observational 
value  of  the  mean  error,  in  distinction  from  the  theoretical  value, 
e,  which  is  a  fixed  function  of  h. 

Thus  the  equation  asserts  that  the  most  probable  value  of  h 
is  found  by  assuming  the  theoretical  value  of  thf  mean  error  to 
be  the  same  as  its  observational  value.  In  other  words,  it  is  a 
consequence  of  the  accepted  law  of  facility  that  the  measure  of 
precision  of  a  set  of  observations  equally  good  is  proportional 
to  the  reciprocal  of  the  mean  error  as  determined  from  the 
observations  themselves. 


§V.]        MEAN  AND  PROBABLE  ERRORS.          55 

Formula  for  the  Mean  and  Probable  Errors. 

71.  The  quantity  2p(x  —  <z)2  in  the  value  of  h,  equation  (2), 
Art.  69,  is  the  sum  of  the  weighted  squares  of  the  actual  errors 
of  the  observed  values  xv  ,  x2  ,  .  .  .  xn  .  Now,  when  a  denotes 
the  weighted  arithmetical  mean,  x—  a  must  be  replaced  by  v  +  dt 
as  in  Art.  67,  and 

(i) 


The  value  of  d,  which  is  the  error  of  the  arithmetical  mean,  is 
of  course  unknown  ;  it  may  be  either  positive  or  negative,  but, 
since  £2  is  essentially  positive,  the  true  value  of  2p(x  —  aj" 
always  exceeds  2pv*.  The  best  correction  we  can  apply  to  the 
approximate  value  2'pz?  is  found  by  giving  to  <52  in  equation  (i) 
its  mean  value  ;  for,  by  adopting  this  as  a  general  rule  we  shall 
commit  the  least  error  in  the  long  run.  Now  we  have  seen  in 
Art.  67  that  d  follows  a  law  of  probability  of  the  usual  form  in 
which  the  measure  of  precision  is  h  ^  Ip,  hence  the  mean  value 
of  <52  is  the  same  as  the  mean  squared  error  found  in  Art.  50, 
except  that  h  is  changed  to  h  V  2p.  That  is  to  say,  the  mean 
value  of  <52  is 

i 


Putting  this  in  place  of  <53  in  equation  (i)  we  have 


,     =  .      .. 

Equation  (2),  Art.  69,  may  be  written  in  the  form 

I  =  2IX*  -  «)', 

and,  employing  the  value  just  determined,  we  have 


whence  we  derive 

h  =  iJ  Tvv^ (3) 


56  THE  COMBINATION  OF  OBSERVATIONS.       [Art.  71 

for  the  most  probable  value  of  h  for  an  observation  of  weight 
unity. 

72.  The  resulting  value  of  the  mean  error  of  an  observation 
whose  weight  is  unity  is 


and  by  Art.  68,  the  mean  error  of  the  arithmetical  mean  whose 
weight  is  2p  is 


Again,  the  value  of  the  probable  error  of  an  observation 
whose  weight  is  unity  is 


and  that  of  the  weighted  arithmetical  mean  is 


The  constant  0.6745  is  the  reciprocal  of  that  which  occurs  in 
equation  (2),  Art.  51. 

For  a  set  of  equally  good  observations  we  have,  by  putting 
pi  =p*  =  ...  =pn  =  i, 


»•  =  0.6745  -VJT        ......    (5) 

for  the  probable  error  of  a  single  observation,  and 


for  the  probable  error  of  the  simple  arithmetical  mean. 


§V.]  VALUE  OF  h  IN  TARGET  PRACTICE.  57 

The  Most  Probable  Value  of  h  in  Target  Practice. 

73.  We  have  seen  in  Art.  55  that  in  target  practice  the  prob- 
ability of  hitting  an  elementary  area  «,  situated  at  the  distance  r 
from  the  point  aimed  at,  is 


Suppose  that  n  shots  have  been  made,  the  first  falling  upon 
the  area  at  ,  the  second  upon  a2  ,  and  so  on  ;  then,  before  the 
shots  were  made,  the  probability  that  the  shots  should  fall  upon 
these  areas  in  the  given  succession  is 


a,  ...on. 

Hence,  the  shots  having  been  made,  the  probabilities  of  different 
values  of  h  are  proportional  to  the  values  they  give  to  the 
expression 


Making  this  function  of  h  a  maximum,  we  have 

e-»*"l2nh*n-l-2kn+lZr>]=o, 
whence  we  have,  for  the  most  probable  value  of  h, 


=  Vl& 


the  value  quoted  in  Art.  61. 
74.  The  value  of  e2  hence  derived  is 


2n  2n 


where  e  is  the  mean  error  for  the  component  deviations,  which 
are  the  values  of  x  and  y  respectively.  The  values  of  e2  as 
determined  from  the  lateral  and  vertical  deviations  respectively, 
are 


THE  COMBINATION  OF  OBSERVATIONS.      [Art.  74 


Thus  the  value  of  e2,  which  we  have  derived  from  the  total 
deviations,  or  values  of  r,  is  the  mean  of  its  most  probable 
values  as  separately  derived  from  the  two  classes  of  component 
deviations. 

It  will  be  noticed  that  neither  of  the  quantities  2V,  2j/2  or 
2V2  needs  to  be  corrected  as  in  Art.  71,  because  we  are  here 
dealing  with  actual  errors  and  not  with  residuals.* 

The  Computation  of  the  Probable  Error. 

75.  The  annexed  table  gives  an  example  of  the  application 
of  formulae  (5)  and  (6),  Art.  72.  The  seventeen  values  of  x  in 


X 

V 

V 

4-524 

+  .0185 

.00034225 

4-500 

-  -0055 

3025 

4-515 

+  .0095 

9025 

4-508 

+  .0025 

625 

4-513 

+  .0075 

5625 

4-5i  i 

+  .0055 

3025 

4-497 

-.0085 

7225 

4.507 

+  .0015 

225 

4.501 

--0045 

2025 

4.502 

--0035 

1225 

4485 

—  .0205 

42025 

4.519 

+  .0135 

18225 

4.517 

+  -0115 

13225 

4.504 

—  .0015 

225 

4-493 

—  .0125 

15625 

4.492 

-  -0135 

18225 

4.505 

—  .0005 

25 

a  =  4-5°5i81! 

r  =  4-5°55 

2V  =  .00173825 

*If  the  position  of  the  point  aimed  at  had  been  inferred  from  the 
shot  marks,  as  in  example  22  of  the  preceding  section,  it  would  have 
been  necessary  to  change  n  into  n  —  i,  as  in  the  case  of  errors  of  obser- 
vation. So  also  this  change  should  be  made  when  the  errors  employed 
are  measured  from  the  mean  point  of  impact,  as  in  testing  pieces  of 
ordnance. 


§V.]        COMPUTATION  OF  THE  PROBABLE  ERROR.  59 

the  first  column  are  independent  measurements  of  the  same 
quantity  made  by  Prof.  Rowland  for  the  purpose  of  determining 
a  certain  wave  length.  At  the  foot  of  the  column  is  the  arith- 
metical mean  of  the  seventeen  observations.  The  second  column 
contains  the  residuals  found  by  subtracting  this  from  the  separate 
observations.  The  values  of  i?  in  the  third  column  are  taken 
from  a  table  of  squares,  and  their  sum  is  written  at  the  foot  of 
the  column.  Dividing  this  by  16,  the  value  of  n  —  I,  we  find 

V     2 

*"_     =  0.00010864, 

and  taking  the  square  root, 

e  =0.01042. 
Multiplying  by  the  constant  0.6745  we  h&ve 

r  =  0.00703 

for  the  probable  error  of  a  single  observation. 
Again,  dividing  by  ij  17,  we  have 

r0  =  0.00171 

for  the  probable  error  of  the  final  determination,  which  may 
therefore  be  written 

x  =  4.5055  ±  0.0017. 

It  will  be  noticed  that  nine  of  the  residuals  are  numerically 
less  and  eight  are  numerically  greater  than  the  value  we  have 
found  for  the  probable  error  of  a  single  observation. 

76.  The  equation 


derived  in  Art.  43,  enables  us  to  abridge  somewhat  the  com- 
putation of  2V,  and  to  reduce  the  extent  to  which  a  table  of 
squares  is  needed.  Thus,  if  we  use  the  value  of  a  to  three 
places  of  decimals,  namely  a  =  4.505,  in  forming  the  values  of 


6O  THE  COMBINATION  OF  OBSERVATIONS.       [Art.  -6 

v,  each  of  these  quantities  will  be  algebraically  greater  than  it 
should  be  by  T8T  of  a  unit  in  the  third  decimal  place.     Putting 


hence  2V,  as  found  on  this  supposition,  will  be  too  great  by 
3ff-  of  a  unit  in  the  sixth  decimal  place.  The  columns  headed 
v  and  #2  would  then  stand  as  follows  : 

v  v> 

+  .019  .000361 

-  -005  25 

+  .010  100 

+  -003  9 

+  .008  64 

+  .006  36 

—  .008  64 

+  .002  4 

-  .004  1  6 

-  .003  9 

—  .020  400 
+  .014  196 
+  .012  144 

—  .001  i 

—  .012  144 

—  .013  169 
.000  O 

—  (v  +  »)~  =  .001742 

and  making  the  correction  found  above,  we  have 
Zv*  =  .001738^, 

which  is  the  exact  value. 

The  smallness  of  the  correction  is  due  to  the  fact  that  JV  is 
a  minimum  value.  The  correction  might  have  been  neglected, 
being,  in  this  case,  only  about  -fa  of  the  correction  made  in  the 
formula  on  account  of  the  mean  value  of  the  unknown  error  in 
the  arithmetical  mean. 


§V.]         COMPUTATION  OF  THE  PROBABLE  ERROR.  6 1 

77.  As  an  example  of  the  application  of  the  formulae  involving 
weights,  let  us  suppose  that  instead  of  the  seventeen  observations 
in  the  preceding  article  we  were  given  only  the  means  of  certain 
gr  oups  into  which  the  seventeen  observations  may  be  separated. 
These  means  we  have  seen  may  be  regarded  as  observations 
having  weights"  equal  to  the  respective  numbers  of  observations 
from  which  they  are  derived.  The  annexed  table  presents  the 

p      x        v         v* 

2  4.512     +  .0065    .00004225    .00008450 

1  4.515     +.0095       9025        9025 
4     4.507      +  .0015        225         900 

3  4-503      -.0025        625        1875 

2  4.502  -.0035  1225  2450 

2  4.5II  +.0055  3025  6050 

3  4497  -.0085  7225  21675 
a  =  4.5055                                             Ipv*  =  .00050425 

data  in  such  a  form,  the  first  value  of  x  being  the  mean  of  the 
first  two  values  in  the  preceding  table,  the  next  being  the  third 
observation,  the  next  the  mean  of  the  following  four,  and  so  on. 
The  weighted  mean  of  the  present  seven  values  of  x  of  course 
agrees  with  the  final  value  before  found.  The  values  of  v  and 
of  v1  are  formed  as  before,  and  the  values  oipvz  are  given  in  the 
last  column,  at  the  foot  of  which  is  the  value  of  Ipvz .  Dividing 
this  by  6,  the  present  value  ofn—  i,  we  find 

™    =  0.00008304, 

and,  multiplying  the  square  root  of  this  by  0.6745,  the  value 
of  the  probable  error  of  an  observation  whose  weight  is  unity  is 

r  =0.00615. 

The  probable  error  of  the  weighted  mean  found  by  dividing 
this  by  tj  17,  the  value  of  *J  2p>  is 

r0  =  0.00149. 


62  THE  COMB  IN  A  TION  OF  OBSER  VA  TIONS.       [Art.  78 

78.  The  value  of  r  found  above   corresponds  to  a  single 
observation  of  the  set  given  in  Art.  75.     It  differs  considerably 
from  the  value  found  in  that  article.     The  discrepancy  is  due  to 
the  fact  that  in  Art.  76  we  did  not  use  all  the  data  given  in  Art. 

75,  and  it  is  not  to  be  expected  that  the  most  probable  value  of 
h  which  can  be  deduced  from  the  imperfect  data  should  agree 
with  that  deduced  from  the  more  complete  data.     In  one  case 
we  have  seventeen  discrepancies  from  the  arithmetical  mean, 
due  to  accidental  errors,  upon  which  to  base  an  estimate  of  the 
precision  of  the  observations ;  in  the  other  case  we  have  but 
seven  discrepancies.      The  result  in    the  former  case  is  of 
course  more  trustworthy  ;  and  in  general,  the  larger  the  value 
of  n,  the  more  confidence  can  we  place  in  our  estimate  of  the 
measures  of  precision. 

79.  It  should  be  noticed  particularly  that  the  weighted  obser- 
vations in  Art.  76  are  not  equivalent  to  a  set  of  seventeen 
observations  of  which  two  are  equal  to  the  first  value  of  x,  one 
to  the  second,  four  to  the  third,  and  so  on,  except  in  the  sense  of 
giving  the  same  'mean  value.     Compare  Art.  10.     Such  a  set 
would  exhibit  discrepancies  very  much  smaller  on  the  whole  than 
those  of  the  seventeen  observations  in  Art.  75.    Accordingly, 
the  value  of  e2  in  the  supposed  case  would  be  very  much  smaller 
than   that  found  above  for  the  weighted  observations.     The 
value  of  Zv*  would  in  fact  be  the  same  as  that  of  2pv*  in  Art. 

76,  but  it  would  be  divided  by  16  instead  of  by  6. 

The  approximate  equality  of  the  results  in  Art.  75  and  Art.  76 
is  due  to  the  fact  that  the  z>2's,  of  which  seventeen  exist  in  each 
sum,  are  on  the  average  very  much  diminished*  when  the  mean 
of  a  group  is  substituted  for  the  separate  observations,  and  this 

*  The  amount  of  this  diminution  is,  however,  largely  a  matter  of  chance. 
For  example,  if  we  had  taken  the  seven  groups  in  such  a  manner  that 
the  successive  values  of/  were  2,  3,  2,  4,  2,  i,  3,  we  should  have  found 

r  =  0.00833, 

differing  in  excess  from  that  of  Art.  75  still  more  than  that  obtained 
above  does  in  defect. 


§  V.]        COMPUTA  TION  OF  THE  PROBABLE  ERROR.          63 

makes  up  for  the  change  in  the  denominator  by  the  decrease  in 
the  value  of  n. 

80.  Different  weights  are  frequently  assigned  to  observations 
made  under  different  circumstances,  according  to  the  judgment 
of  the  observer.     Thus  an  astronomer  may  regard  an  observa 
tion  made  when  the  atmosphere  is  exceptionally  clear  as  worth 
two  of  those  made  under  ordinary  circumstances.     Regarding 
the  latter  as  standard  observations  having  the  weight  unity,  he 
will  then  assign  the  weight  2  to  the  former.     As  explained  in 
the  preceding  article  this  is  not  equivalent  to  recording  two 
standard  observations,  each  giving  the  observed  value.     The 
latter  procedure  would  lead  to  an  erroneous  estimate  of  the 
degree  of  accuracy  attained. 

The  Values  of  h  and  r  derived  from  the  Mean  Absolute 
Error. 

81.  The  mean  absolute  error  >?  is  a  fixed  function  of  hy  viz: 


hence,  if  we  were  able  to  determine  it  independently,  we  should 
have  a  means  of  finding  the  value  of  h,  and  consequently  that 
ofr. 

In  the  case  of  n  equally  good  observations,  let  \_x  —  a]  denote 
the  numerical  value  of  an  error  taken  as  positive,  then 


w 


is  the  arithmetical  mean  of  the  absolute  values  of  the  n  actual 
errors.  This  may  be  called  the  observational  value  of  the  mean 
absolute  error  in  distinction  from  the  theoretic  value  given  in 
equation  (i),  which  is  the  value  of  this  mean  in  accordance  with 
the  law  of  probability,  when  the  measure  of  precision  is  h. 
If  we  assume  these  values  to  be  equal,  we  obtain 


n 


64  THE  COMBINATION  OF  OBSERVATIONS.      [Art.  Si 


whence 


and 


If  in  this  formula  we  put  for  a  the  arithmetical  mean,  so 
that  2[x  —  a]  becomes  2[v],  it  gives  the  apparent  probable 
error,  that  is,  the  value  r  would  have  if  the  arithmetical  mean 
were  known  to  be  the  true  value  of  x.  Denoting  this  by  r1 ', 
we  have  then 

^  =  ,^32  =  0.8453^.      ...     (5) 

82.  It  is  obvious  from  Arts.  71  and  72  that  the  values  of  r' 
and  r  as  derived  from  the  square  of  the  residuals  are 

r'  =  0.6745  Y-J~-  ,        r  =  0.6745  fA—  , 

so  that 

r:r'  =  j/n:tf(n-i)* (6) 

*  This  relation  between  the  apparent  and  the  real  probable  error  is  de- 
rived directly  by  C.  A.  F.  Peters  (Berliner  Astronomisches  Nachrichten, 
1856,  vol.  xliv.  p.  29)  as  follows:  If  et ,  <?3 ,  .  .  .  en  are  the  true  errors, 
that  of  the  arithmetical  mean  is 


then 


i 

en ,     etc. 

n 


Since  r  is  the  probable  error  of  each  e,  and  r1  that  of  each  v,  the  formula 
for  the  probable  error  of  a  linear  function  of  independent  quantities  (see 
Art.  89)  gives 


This  result  is  used  by  Peters  to  establish  the  formula  derived  above,  but 
it  may  also  be  used  in  place  of  the  method  of  Art.  71  for  the  correction  of 
the  apparent  value  of  r  in  terms  of  *£v2. 


§  V.]  FORMULA  INVOLVING  MEAN  ABSOLUTE  ERROR.  6$ 


Combining  this  result  with  equation  (5)  we  have 

•^ir     -| 


and  hence,  for  the  probable  error  of  the  arithmetical  mean, 

......    00 


As  an  illustration,  let  us  apply  these  formulae  to  the  observa- 
tions given  in  Art.  75,  for  which  we  find  2[v]  =  0.1405.  Sub- 
stituting this  value,  and  putting  n  =  17,  we  find 

r  =  0.00720,         r0  =  0.00175. 

These  values  agree  closely  with  those  derived  in  Art.  75 
from  the  formulae  involving  2v*,  which  indeed  give  the  most 
probable  values  of  r  and  r0  ,  but  involve  much  more  numerical 
work,  especially  when  n  is  large. 

83.  In  order  to  adapt  the  formulae  of  Art.  82  to  the  case  of 
weighted  observations,  it  is  necessary  to  reduce  the  errors  to 
the  same  scale;  in  other  words,  to  make  them  proportional  to 
the  reduced  errors  or  values  of  /,  see  Art.  47.  Since  the 
measures  of  precision  are  proportional  to  the  square  roots  of 
the  weights,  this  is  effected  by  multiplying  each  error  by  the 
square  root  of  the  corresponding  weight.  The  products  may 
be  regarded  as  errors  belonging  to  the  same  system,  namely, 
that  which  corresponds  to  the  weight  unity. 

Hence  equation  (7)  gives  for  the  probable  error  of  an  obser- 
vation whose  weight  is  unity 


and  for  the  probable  error  of  the  weighted  arithmetical  mean 
we  have 


66  THE  COMBINATION  OF  OBSERVATIONS.       [Art.  $3 

Examples. 

1.  A  line  is  measured  five  times  and  the  probable  error  of 
the  mean  is  .016  of  a  foot.     How  many  additional  measure- 
ments of  the  same  precision  are  required  in  order  to  reduce  the 
probable  error  of  the  determination  to  .004  of  a  foot?         75. 

2.  It  is  required  to  determine  an  angle  with  a  probable  error 
less  than  o".25.     The  mean  of  twenty  measurements  gives  a 
probable  error  of  0^.38 ;  how  many  additional  measurements 
are  necessary  ?  27. 

3.  If  the  probable  error  of  each  of  two  like  measurements  of  a 
foot  bar  is  .00477  of  an  inch,  what  is  the  probable  error  of  their 
mean  ?  .00337. 

4.  Ten  measurements  of  the  density  of  a  body  made  with 
equal  precision  gave  the  following  results : 

9.662,  9.664,  9.677,  9.663,  9.645, 

9.673,  9.659,  9.662,  9.680,  9.654. 

What  is  the  probable  value  of  the  density  of  the  body  and  the 
probable  error  of  that  value  ?  9.6639  ±  .0022. 

5.  Forty  micrometric  measurements  of  the  error  of  position 
of  a  division  line  upon  a  standard  scale  gave  the  following 
results : 


3.68 

5.08 

2.8  1 

4-43 

548 

4.21 

3-28 

5-21 

3-u 

2-95 

4-65 

3-43 

3-76 

5-23 

3.78 

443 

4.76 

6-35 

3-27 

3-26 

4-59 

445 

3.22 

2.28 

2-75 

3.78 

4.08 

2.48 

2.64 

3-95 

3-98 

4.10 

4.15 

449 

4.5i 

4.84 

2.98 

2.66 

3.9i 

4.18 

Find  the  probable  value  of  the  quantity  measured  and  its  prob- 
able error.  3-93°  ±  0.097. 

6.  In  the  preceding  example  what  is  the  probable  error  of  a 
single  observed  quantity:  i°,  by  the  formula  involving  the 
squares  of  the  errors;  2°,  by  that  involving  the  absolute  errors? 

i°,  r  =  0.616 ;  2°,  r  =  0.618. 


§  V.]  EXAMPLES.  67 

7.  An  angle  in  the  primary  triangulation  of  the  U.  S.  Coast 
irvey  was   measured  twenty-four  times  with   the   following 

1 A 


Survey  was 
results 


44  -45 

49-20 

51-05 

5J-75 

51-05 

49-25 

50  -55 

48.85 

47-85 

49.00 

51-70 

46.75 

50  -95 

4740 

50.  6o 

52.35 

49-05 

49-25 

48  .90 

47-75 

48.45 

51.30 

50.55 

5340 

Find  the  probable  error  of  a  single  measurement,  and  the  final 
determination  of  the  angle.  i"-35 :  1 16°  43'  49".64  ±  o".28. 

8;  In  example  7,  taking  the  means  of  the  six  groups  of  four 
observations  each,  determine  the  probable  error  of  the  first  of 
these  means :  i°,  considered  as  a  measurement  of  four  times  the 
weight  of  those  in  example  7  ;  2°,  directly  as  one  of  six  obser- 
vations of  equal  weight ;  3°,  as  a  determination  from  its  four 
constituents.  i°,  0^.67  ;  2°,  0^.72 ;  3°,  i".oo. 

9.  An  interval  of  600  units  as  determined  by  a  micrometer 
was  forty  times  measured  to  determine  the  error  in  the  pitch  of 
the  screw,  with  the  following  results : 


600.0 

604.8 

600.7 

601.4 

602.0 

602.6 

600.0 

602.4 

599.7 

606.  1 

602.4 

603.4 

602.7 

602.7 

600.7 

602.4 

599.5 

604.7 

601.6 

603.1 

603.7 

600.9 

601.4 

602.  1 

604.6 

602.  1 

601.7 

601.8 

602.  1 

601.4 

602.9 

603.6 

603.9 

602.2 

601.4 

600.6 

602.3 

600.8 

602.9 

603.6 

Find  the  probable  value  of  the  interval  and  its  probable  error. 

602. 22  ±0.157. 


VI. 

THE  FACILITY  OF  ERROR  IN  A  FUNCTION  OF  ONE  OR  MORE 
OBSERVED  QUANTITIES. 

The  Linear  Function  of  a  Single  Observed  Quantity. 

84.  If  the  value  of  an  observed  quantity  X  be  subject  to  an 
error  .*,  the  value  of  a  given  function  of  X,  say  Z=f  (X),  will 
be  subject  to  a  corresponding  error  2.  Assuming  x  to  follow 
the  usual  law  of  facility,  h  being  the  measure  of  precision  and  r 
the  probable  error,  we  have  now  to  determine  the  law  of  facility 
of  2)  for  any  form  of  the  function/. 

Let  us  first  consider  the  linear  function 

Z=mX+b, 

where  m  and  b  are  constants.  The  case  is  obviously  the  same 
as  that  of  the  simple  multiple  mX,  the  relation  between  the 
corresponding  errors  being 

z  =.  mx. 

The  probability  that  the  error  z  falls  between  z  and  z  +  dz  is 
the  £  ime  as  the  probability  that  x  falls  between  x  and  x  +  dxt 
namely, 


Expressing  this  in  terms  of  z,  it  becomes 


or,  putting  -  =  H, 


Thus  the  law  of  facility  for  Z  is  of  the  same  form  as  that  for  Xt 


§VL]  FUNCTIONS  OF  A  SINGLE  QUANTITY.  69 

the  measure  of  precision  being  found  by  dividing  that  ofXby  m  ; 
and,  denoting  the  probable  error  of  Z  by  R,  we  -have  (since 
probable  errors  are  inversely  as  the  measures  of  precision) 

R  =  mr, 

and  the  same  relation  holds  between  either  of  the  other  measures 
of  the  risk  of  error. 

The  curves  of  facility  for  X  and  Z  are  related  in  the  same 
manner  as  those  drawn  in  Fig.  4,  page  30,  and  the  process  of 
passing  from  one  to  the  other  is  that  described  in  Art.  46;  that 
is  to  say,  the  abscissas  which  represent  the  errors  are  multiplied 
by  m,  and  then  the  ordinates  are  divided  by  m,  so  that  the  areas 
standing  upon  the  corresponding  bases  dx  and  dz  shall  remain 
equal. 

Non-Linear  Functions  of  a  Single  Observed  Quantity. 

85.  A  non-linear  function  of  an  observed  quantity  subject  to 
the  usual  law  of  facility  does  not  strictly  follow  a  law  of  facility 
of  the  same  form.  If,  however,  as  is  usually  the  case,  the  error 
x  is  very  small,  any  function  of  the  observed  quantity  will  very 
nearly  follow  a  law  of  the  usual  form.  Let  a  be  the  'true  value 
of  the  observed  quantity,  then 

X=a  +  x, 
and 


Expanding  by  Taylor's  Theorem,  and  neglecting  the  higher 
powers  of  x*  we  may  take 

Z=f(a)  +  Xf'(a-), 

which  is  of  the  linear  form.     Hence  we  may  regard  Zas  subject 
to  the  usual  law  of  facility,  its  probable  error  being 

R=rf(*), 

or,  putting  the  observed  value  in  place  of  0, 


*The  ratio  of  the  square  of  the  error  to  the  error  itself  is  the  vwiue  of 
the  error  considered  as  a  number,  and  it  is  this  numerical  value  which 
must  be  small. 


THE  FACILITY  OF  ERROR  IN  A  FUNCTION.     [Art.  86 


The  Facility  of  Error  in  the  Sum  or  Difference  of  Two 
Observed  Quantities. 

86.  Let  X  and  Y  be  two  observed  quantities  subject  to  the 
usual  law  of  facility  of  error,  their  measures  of  precision  being  h 
and  k  respectively.  If 

Z=X+Y, 

the  relation  between  the  errors  of  Z,  X  and  Y  is  obviously 

z  =  x  +y. 

In  order  to  find  the  facility  of  £,  that  is,  the  probability  that  2 
shall  fall  between  z  and  z  +  dz,  let  us  first  suppose  that  x  has  a 
definite  fixed  value.  With  this  hypothesis,  the  probability  in 
question  is  the  same  as  the  probability  that^/  shall  fall  between 
>/andj/+  dy,  where 

y  —  z  —  x,      and      dy  =  dz. 
This  probability  is 

A  e-^dy,       or      A  '-**-***• 

VTT  V* 

Multiplying  by  the  elementary  probability  of  the  hypothesis 
made,  which  is 

4-  e-"+dx, 

V* 

we  have 


for  the  probability  that  the  required  event  (namely,  the  occur- 
rence of  the  particular  value  of  z)  shall  happen  in  this  particular 
way,  that  is,  in  connexion  with  the  particular  value  of  x.  To 
find  the  total  probability  of  the  event  we  therefore  sum  the 
above  expression  for  all  possible  values  of  xy  thus  obtaining 

...      (2) 


§VI.]         THE  SUM  OF  TWO  OBSERVED  QUANTITIES.        *J\ 
The  exponent  of  e  in  this  expression  may  be  written 


whence,  putting  a  =  and 

/P  =  £2  - 

the  expression  (2)  becomes 

,f "  e 


Since  a  is  independent  of  ^,  the  value  of  the  integral  contained 

in  this  expression  is,  by  Art.  39,    ,  ,£  '.    ,^  ;  hence  the  proba- 

Y    ft   +  K  ) 


bility  that  z  shall  fall  between  z  and  z  +  dzis 

,  H      — 

'  # 


87.  The  result  just  obtained  shows  that  the  sum  of  two 
quantities  subject  to  the  usual  law  of  facility  of  error  is  subject 
to  a  law  of  the  same  form,  its  measure  of  precision  being  deter- 
mined by  equation  (3). 

Writing  equation  (3)  in  the  form 


it  is  evident  that,  if  rlt  r*  and  R  be  the  probable  errors  of  X,  Y 
and  X  +  Y,  we  shall  have 

X^rl  +  rt, 

the  same  relation  holding  in  the  case  of  either  of  the  other 
measures  of  risk  of  error. 
For  the  difference 

Z=X-Y, 

we  have  the  same  result;  for  the  errors  of—  Khave  obviously 
the  same  law  of  facility  as  those  of  K 


72         THE  FACILITY  OF  ERROR  IN  A  FUNCTION.      [Art.  88 

88.  As  an  illustration,  suppose  the  latitude  y  and  the  polar 
distance  p  of  a  circumpolar  star  to  be  determined  from  the 
altitudes  of  the  star  at  its  upper  and  lower  culminations.  Since 

hi  =  <P+p      and      bz  =  <f>—p, 
we  have 


Then,  r^  and  r2  denoting  the  probable  errors  of  hl  and  h^  respec- 
tively, that  of  h^  +  hi  and  also  that  of  ^  —  h*  is  V  0"?  +  *1),  hence 
the  probable  error  both  of  <p  and  of/  when  thus  determined  is 


Linear  Function  of  Several  Observed  Quantities. 
89,  It  follows  from  Arts.  84  and  87  that  the  linear  function 

Z=b  +  m1Xl  +  m2X2  +  .  .  .  4-  w»^,  .     .     .     .     (i) 

of  «  observed  quantities  is  subject  to  the  usual  law  of  facility,* 
its  probable  error  being 

jR=  *J(mir[  +  mlrl  +  ...  +  mlr%,      ...    (2) 

where  rly  ra,  .  .  ..  r%  are  the  probable  errors  of  the  several 
observed  quantities. 

In  particular,  if  the  n  quantities  have  the  same  probable  error 
r,  the  probable  error  of  their  sum  is  r»Jn.     The  probable  error 

of  their  arithmetical  mean,  which  is  —  of  this  sum,  is  therefore 


-T—  .     This  result  agrees  with  that  found  in  Art.  64,  where, 


*  The  fact  that  the  law  of  facility  thus  reproduces  itself  has  often  been 
regarded  as  confirmatory  of  its  truth.  This  property  of  the  Iaw^~ft8xt 
results  from  its  being  a  limiting  form  for  the  facility  of  error  in  the  linear 
function  Z,  when  n  is  large,  whatever  be  the  forms  of  the  facility  functions 
for  Xlt  X^,  .  .  .  Xn,  Compare  the  foot-note  on  page  49,  and  see  the 
memoir  there  referred  to.  It  follows  that  "we  shall  obtain  the  same 
law  tf-feaxa  (for  a  single  observed  quantity)  if  we  regard  each  actual 
eiror  as  formed  by  the  linear  combination  of  a  large  number  of  errors 
cue  to  different  independent  sources." 


§VL]  NON-LINEAR  FUNCTIONS.  73 

however,  the  n  quantities  were  all  observed  values  of  the  same 
quantity,  and  the  arithmetical  mean  was  under  consideration  by- 
virtue  of  its  being  the  most  probable  value  in  accordance  with 
the  law  of  facility. 

90.  It  is  to  be  noticed  that  in  formula  (2)  it  is  essential  that 
the  probable  errors  rlt  r2,  .  .  .  rn  should  be  the  results  of  inde- 
pendent determinations.     For  example,  in  the  illustration  given 
in  Art.  88,  we  have  h^  =  <f>  +  p,  whence  we  should  expect  to  find 

(prob.  err.  of  fitf  =  (prob.  err.  of  ^)2  +  (prob.  err.  of  />)'  ; 

but  it  will  be  found  that  this  is  not  true  when  the  probable 
errors  of  <p  and  of  p  are  determined  as  in  that  article.  In  fact, 
in  the  demonstration  given  in  Art.  86,  it  is  assumed  that  the  law 
of  facility  for  Y  holds  true  when  X  has  a  definite  fixed  value  ; 
but  in  the  present  illustration  the  law  of  facility  found  for  y  does 
not  hold  true  for  a  definite  fixed  value  of/.* 

The  Non-  Linear  Function  of  Several  Observed  Quantities. 

91.  Supposing,  as  in  Art.  85,  that  the  errors  of  the  observed 
quantities  are  small  compared  to  the  quantities  themselves, 
we  may  replace  any  function  by  an  approximately  equivalent 
function  of  a  linear  form.     Thus,  denoting  the  true  values  of  the 
observed  quantities  Xl  ,  Xt  ,  .  .  .  Xn  by  al  ,  az  ,  .  .  .  an  ,  we  have 


tt...         =«!      xlt  a,      *f,  .  .  .  an      *„. 

Expanding,  and  neglecting  powers  and  products  of  the  small 
quantities  xlt  xtj  .  .  .  xnt  we  obtain  the  approximate  value 


which  is  of  the  linear  form.  Hence,  in  accordance  with  equation 
(2),  Art.  89,  the  probable  error  of  Z  may  be  determined  by  the 
equation 


*  If  the  value  of  /  were  known,  each  value  of  hi  would  imply  a  special 
value  of  //a,  and  therefore  the  probability  of  ^  would  no  longer  be  that 
found  in  Art.  88. 


74        THE  FACILITY  OF  ERROR  IN  A  FUNCTION.       [Art.  91 

Examples* 

1.  If  the  probable  error  in  measuring  the  radius  a  of  a  circle 
is  r,  what  are  the  probable  errors  of  the  circumference  and  of 
the  area  ?  zxr  ;  2xar. 

2.  What  is  the  probable  error  of  Iog10#,  r  being  the  probable 

error  of  x  ?  <M343  — 

x 

3.  If  measurements  of  adjacent  sides  of  a  rectangle  give  a  ±  r± 
and  b  ±  ra  ,  what  is  the  probable  error  of  the  area  ab  ? 


4.  If  the  rectangle  is  found  to  be  a  square  and  the  sides  are 
measured  with  the  same  precision,  show  that  the  probable  error 
of  the  area  is  the  same  as  if  it  were  known  to  be  a  square  ;  but 
if  r±  and  r^  are  not  equal,  the  area  is  obtained  with  less  accuracy 
than  it  would  be  if  it  were  known  to  be  a  square. 

5.  An  angle  observation  is  the  difference  between  two  read- 
ings of  the  limb  of  the  instrument  ;  if  r  is  the  probable  error  of 

the  angle,  what  is  the  probable  error  of  each  reading?    _/" 

4/2* 

6.  The  zenith  distance  of  a  star  observed  in  the  meridian  is 

C  =  21°  if  2o".3,  with  the  mean  error  2^.3, 
and  the  declination  of  the  star  is  given 

d  =  19°  30'  I4".8,  with  the  mean  error  o".8  : 

what  is  the  mean  error  of  the  latitude  of  the  place  of  observation 
found  from  the  formula  <p  =  £  +  8  ? 

y>  —  40°  47'  35".!,  with  the  mean  error  2".44. 

7.  The  latitude  of  a  place  has  been  found  with  the  mean  error 
o".25,and  the  meridian  zenith  distance  of  stars  observed  at  that 
place  with  a  certain  instrument  has  been  found  to  be  subject  t<> 
the  mean  error  o".62  ;  what  is  the  mean  error  of  the  declinations 
of  the  stars  deduced  by  the  formula  3  =  <p  —  C?  o".67. 

8.  The  correction  of  a  chronometer  is  found  to  be  +  i2m  I38.2, 
with  the  mean  error  os.3  ;  ten  days  later  the  correction  is  found 
to  be  +  I2m  2I8.4,  with  the  same  mean  error;  what  is  the  mean 
daily  rate  and  its  mean  error?  4-  O*.82  ;  o'.O42. 


§VL]  EXAMPLES.  75 

9.  If  the  error  of  a  single  measurement  of  an  angle  by  a 
repeating  circle  consists  of  parts  due  to  sighting  and  reading 
respectively,  so  that 

show  that  the  probable  error  when  the  angle  is  repeated  n  times 
is 


10.  If  the  measured  sides  of  a  rectangle  have  the  same  prob- 
able error,  show  that  the  diagonal  is  determined  with  the  same 
precision  as  either  side. 

11.  The  compression  of  the  earth's  meridian  was  found  to 
be  2^r>  with  a  probable  error  of  .000046 ;  what  is  the  probable 
error  of  the  denominator  294  ?  3.98. 

12.  When  a  line  whose  length  is  /  is  measured  by  the  repeated 
application  of  a  unit  of  measure,  show  that  its  probable  error  is 
of  the  form 

R=r*ll. 

13.  What  is  the  probable  error  of  the  area  of  the  rectangle 
whose  sides  measured  as  in  the  preceding  example  are  z\  and  z*  ? 

rVO^Oi  +  z*)]- 

14.  A  line  of  levels  is  run  in  the  following  manner :  the  back 
and  fore  sights  are  taken  at  distances  of  about  200  feet,  so  that 
there  are  thirteen  stations  per  mile,  and  at  each  sight  the  rod  is 
read  three  times.     If  the  probable  error  of  a  single  reading  is 
0.01  of  a  foot,  what  is  the  probable  error  of  the  difference  of  level 
of  two  points  which  are  ten  miles  apart  ?  .093. 

15.  Show  that  the  probable  error  of  the  weighted  mean  of 
observed  quantities  has  its  least  possible  value  when  the  weights 
are  inversely  proportional  to  the  squares  of  the  probable  errors 
of  the  quantities,  and  that  this  value  is  the  same  as  that  given  in 
Art.  68  for  the  case  of  observed  value  of  the  same  quantity. 


VII. 

THE  COMBINATION  OF  INDEPENDENT  DETERMINATIONS  OF 
THE  SAME  QUANTITY. 

The  Distinction  between  Precision  and  Accuracy. 

92.  We  have  seen  in  Arts.  63  and  67  that  the  final  determi- 
nation of  the  observed  quantity  derived  from  a  set  of  observations 
follows  the  exponential  law  of  the  facility  of  accidental  errors. 
The  discrepancies  of  the  observations  have  given  us  the  means 
of  determining  a  measure  of  the  risk  of  error  in  the  single 
observations,  and  we  have  found  that  the  like  measure  for  the 
final  determination  varies  inversely  as  the  square  root  of  its 
weight  compared  with  that  of  the  single  observation.     Since 
this  weight  increases  directly  with  the  number  of  constituent 
observations,  it  is  thus  possible  to  diminish  the  risk  of  error 
indefinitely;  in  other  words,  to  increase  without  limit  the  pre- 
cision of  our  final  result. 

93.  It  is  important  to  notice,  however,  that  this  is  by  no  means 
the  same  thing  as  to  say  that  it  is  possible  by  multiplying  the 
number  of  observations  to  increase  without  limit  the  accuracy 
of  the  result.     The  precision  of  a  determination  has  to  do  only 
with  the  accidental  errors  ;  so  that  the  diminution  of  the  prob- 
able error,  while  it  indicates  the  reduction  of  the  risk  of  such 
errors,  gives  no  indication  of  the  systematic*  errors  (see  Art.  3) 

*The  term  systematic  is  sometimes  applied  to  errors  produced  by  a 
cause  operating  in  a  systematic  manner  upon  the  several  observations, 
thus  producing  discrepancies  obviously  not  following  the  law  of  accidental 
errors.  Usually  a  discussion  of  these  errors  leads  to  the  discovery  of 
their  cause,  and  ultimately  to  the  corrections  by  means  of  which  they  may 
be  removed.  All  the  remaining  errors,  whose  causes  are  unknown,  are 
generally  spoken  of  as  accidental  errors  ;  but  in  this  book  the  term  acci- 
dental is  applied  only  to  those  errors  which  are  variable  in  the  system  of 
observations  under  consideration,  as  distinguished  from  those  which  have 
aQpmmon,  value,  (or  the  entire  system. 


§  VII.]  PRECISION  AND  ACCURACY.  77 

which  are  produced  by  unknown  causes  affecting  all  the  obser- 
vations of  the  system  to  exactly  the  same  extent. 

The  value  to  which  we  approach  indefinitely  as  the  precision 
of  the  determination  is  increased  has  hitherto  been  spoken  of 
as  the  "true  value,"  but  it  is  more  properly  the  precise  value 
corresponding  to  the  instrument  or  method  of  observation 
employed.  Since  the  systematic  error  is  common  to  the  whole 
system  of  observations,  it  is  evident  that  it  will  enter  into  the 
final  result  unchanged,  no  matter  what  may  be  the  number  of 
observations ;  whereas  the  object  of  increasing  this  number  is 
to  allow  the  accidental  errors  to  destroy  one  another.  Thus  the 
systematic  error  is  the  difference  between  the  precise  value, 
from  which  accidental  errors  are  supposed  to  be  entirely  elimi- 
nated, and  the  accurate  or  true  value  of  the  quantity  sought. 

94.  Hence,  when  in  Art.  64  the  arithmetical  mean  of  n  obser- 
vations was  compared  to  an  observation  made  with  a  more 
precise  instrument,   it  is   important   to  notice   that   this   new 
instrument  must  be  imagined  to  lead  to  the  same  ultimate 
precise  value,  that  is,  it  must  have  the  same  systematic  error  as 
the  actual  instrument,  whereas  in  practice  a  new  instrument 
might  have  a  very  different  systematic  error. 

Again,  in  the  illustration  employed  in  Art.  64,  where  the  final 
determination  of  an  angle  is  given  as  36°  42'. 3  ±  i'.22,  the 
"  true  value,"  which  is  just  as  likely  as  not  to  lie  between  the 
limits  thus  assigned,  is  only  the  true  value  so  far  as  the  instru- 
ment and  method  employed  can  give  it ;  that  is,  the  precise  value 
to  which  the  determination  would  approach  if  its  weight  were 
increased  indefinitely. 

95.  A  failure  to  appreciate  the  distinction   drawn   in  the 
preceding  articles  may  lead  to  a  false  estimate  of  the  value 
of  the   method  of  Least  Squares.     M.  Faye   in   his  "  Cours 
d' Astronomic "  gives  the  following  example  of  the  objections 
which  have   been  urged    against   the    method:    "From   the 
discussion  of  the  transits  of  Venus  observed  in  1761  and  1769; 
M.  Encke  deduced  for  the  parallax  of  the  sun  the  value 

8".57i.i6±o".o370. 


78  INDEPENDENT  DETERMINATIONS.  [Art.  95 

In  accordance  with  this  small  probable  error  it  would  be  a 
wager  of  one  to  one  that  the  true  parallax  is  comprised  between 
8". 53  and  8".6i.  Now  we  know  to-day  that  the  true  parallax 
8". 813  falls  far  outside  of  these  limits.  The  error,  o".24i84,  is 
equal  to  6.536  times  the  probable  error  o".O3y.  We  find  for 
the  probability  of  such  an  error  o.ooooi.  Hence,  adhering  to 
the  probable  error  assigned  by  M.  Encke  to  his  result,  one  could 
wager  a  hundred  thousand  to  one  that  it  is  not  in  error  by 
0.24184,  and  nevertheless  such  is  the  correction  which  we  are 
obliged  to  make  it  undergo." 

Of  course,  as  M.  Faye  remarks,  astronomers  can  now  point 
out  many  of  the  errors  for  which  proper  corrections  were  not 
made  ;  but  the  important  thing  to  notice  is  that,  even  in  Encke's 
time,  the  wagers  cited  above  were  not  authorized  by  the  theory. 
The  value  of  the  parallax  assigned  by  Encke  was  the  most 
probable  with  the  evidence  then  known,  and  it  was  an  even  wager 
that  the  complete  elimination  of  errors  of  the  kind  that  produced 
the  discrepancies  or  contradictions  among  the  observations  could 
not  carry  the  result  beyond  the  limit  assigned ;  but  the  existence 
of  other  unknown  causes  of  error  and  the  probable  amount  of 
inaccuracy  resulting  from  them  is  quite  a  different  question. 

Relative  Accidental  and  Systematic  Errors. 

96.  Let  us  now  suppose  that  two  determinations  of  a  quantity 
have  been  made  with  the  same  instrument  and  by  the  same 
method,  so  that  they  have  the  same  systematic  error,  if  any ;  in 
other  words,  they  correspond  to  the  same  precise  value.  The 
difference  between  the  two  results  is  the  algebraic  difference 
between  the  accidental  errors  remaining  in  the  two  determi- 
nations; this  may  be  called  their  relative  accidental  error. 
Regarding  the  two  determinations  as  independent  measure- 
ments of  two  quantities,  if  r\  and >2  are  their  probable  errors, 
that  of  their  difference  is  V  W  +  *1)  I  and>  since  this  difference 
should  be  zero,  the  relative  error  is  an  error  in  a  system  for 
which  the  probable  error  is 
r= 


§VIL]       ACCIDENTAL  AND  SYSTEMATIC  ERRORS.  79 

For  example,  if  the  determination  of  an  angle  mentioned  in  Art. 
94  is  the  mean  often  observations,  it  is  an  even  wager  that  the 
mean  of  ten  more  observations  of  the  same  kind  shall  differ 
from  36°  42\3  by  an  amount  not  exceeding  i'.22  X  V2  or  ir-73- 
Again,  r  being  the  probable  error  of  a  single  observation,  the 

probable  error  of  the  mean  of  n  observations  is  ~-j—  ,   but  the 

discrepancy  from  this  mean  of  a  new  single  observation  is  as 
likely  as  not  to  exceed 


97.  If,  on  the  other  hand,  the  two  determinations  have  been 
made  with   different   instruments   or   by   a   different  method, 
they  may  involve  different  systematic  errors  ;  so  that,  if  each 
determination  were  made  perfectly  precise,  they  would   still 
differ  by  an  amount  equal  to  the  algebraic  difference  of  their 
systematic  errors.     Let  this  difference,  which  may  be  called  the 
relative  systematic  error,  be  denoted  by  8.     Then,  d  denoting 
the  actual  difference  of  the  two  determinations,  while  8  is  the 
difference  between  the  corresponding  precise  values,  we  may 
put 

d=d  +  x, 

in  which  x  is  the  relative  accidental  error. 

The  Relative  Weights  of  Independent  Determinations. 

98.  In  combining  values  to  obtain  a  final  mean  value,  we  have 
hitherto  supposed  their  relative  weights  to  be  known  or  assumed 
beforehand,  as  in  Arts.  75  and  77.     Since  the  squares  of  the 
probable  errors  are  inversely  proportional  to  the  weights,  (Arts. 
66  and  68,)  the  ratios  of  the  probable  errors  both  of  the  con- 
stituents and  of  the  mean  are  thus  known  in  advance,  and  it 

*This  does  not  apply  to  the  residuals  of  the  original  n  observations, 
because  in  taking  a  residual  the  mean  is  not  independent  of  the  single 
observation  with  which  it  is  compared. 


80  INDEPENDENT  DE  TERM  IN  A  TIONS.  [Art.  98 

only  remains  to  determine  a  single  absolute  value  of  a  probable 
error  to  fix  them  all.  In  this  process  it  is  assumed  that  the 
values  have  all  the  same  systematic  error. 

But,  when  the  determinations  are  independently  made,  their 
relative  weights  are  not  known,  and  their  probable  errors  have 
to  be  found  independently.  If  now  it  can  be  assumed  that  the 
s\  stematic  errors  are  the  same,  so  that  there  is  no  relative 
systematic  error,  the  weights  may  be  taken  in  the  inverse  ratio 
of  the  squares  of  the  probable  errors. 

99.  To  determine  whether  the  above  assumption  can  fairly  be 
made  in  the  case  of  two  independent  determinations  whose 
probable  errors  are  r^  and  ?'2,  it  is  necessary  to  compare  the 
difference  d  with  the  relative  probable  error  i/  (r\  +  rl\  Art.  96. 
If  d  is  small  enough  to  be  regarded  as  a  relative  accidental 
error,  it  is  safe  to  make  the  assumption  and  combine  the  deter- 
minations in  the  manner  mentioned  above. 

As  an  example,  let  us  suppose  that  a  certain  angle  has  been 
determined  by  a  theodolite  as 


and  that  a  second  determination  made  with  a  surveyors  transit 

24°  i3'24"±i3".8. 

In  this  case  r±  =  3.  1,  r^  =  13.8  and  d=.  12.  It  is  obvious  that 
a  relative  accidental  error  as  great  as  d  may  reasonably  be 
expected.  (In  fact  the  relative  probable  error  is  14.1  ;  and,  by 
Table  II,  the  chance  that  the  accidental  error  should  be  at  least 
as  great  as  12  is  about  .57.)  We  may  therefore  assume  tha'j 
there  is  no  relative  systematic  error,  and  combine  the  determi- 
nations with  weights  having  the  inverse  ratio  of  the  squares  of 
the  probable  errors.  This  ratio  will  be  found,  in  the  present 
case,  tc  be  about  20  :  i,  and  the  corresponding  weighted  mean 
found  by  adding  ^  of  the  difference  to  the  first  value,  is 

24°  13'  35"43- 
100.  It  appears  doubtful  at  first  that  the  value  given  by  the 


§  VII.]  CONCORDANT  DETERMINATIONS.  8 1 

theodolite  can  be  improved  by  combining  with  it  the  value 
given  by  the  inferior  instrument.  The  propriety  of  the  above 
process  becomes  more  apparent,  however,  if  we  imagine  the 
first  determination  to  be  the  mean  of  twenty  observations  made 
with  the  theodolite;  a  single  one  of  these  observations  will  then 
have  the  same  weight  and  the  same  probable  error  as  the  second 
determination.  Now  the  discrepancy  of  this  new  determination 
from  the  mean  is  such  as  we  may  expect  to  find  in  a  new  single 
observation  with  the  theodolite.  We  are  therefore  justified  in 
treating  it  as  such  an  observation,  and  taking  the  mean  of  the 
twenty-one  supposed  observations  for  our  final  result. 

101.  The  probable  error  of  the  result  found  in  Art.  99  of 
course  corresponds  with  its  weight;  thus,  denoting  it  by  R,  we 
have  R*  =  f  f  r\,  whence  R  =  3^.03,  and  the  final  result  is 

24°  13'  35"-43  ±  3"-o3. 

In  general,  r±  and  rz  being  the  given  probable  errors,  that  of 
the  mean  is  given  by 

«2-^2 

r>2  '\'z 

•ft-    —  — oi — : 5  • 


Determinations  which,  considering  their  probable  errors,  are 
in  sufficient  agreement  to  be  treated  as  in  the  foregoing  articles 
may  be  called  concordant  determinations.  They  correspond  to 
the  same  precise  value  of  the  observed  quantity,  and  the  result 
of  their  combination  is  to  be  regarded  as  a  better  determination 
of  the  same  precise  value. 

The  Combination  of  Discordant  Determinations. 

102.  As  a  second  illustration  of  determinations  independently 
made,  let  us  suppose  that  a  determination  of  the  zenith  distance 
of  a  star  made  at  one  culmination  is 

14°  53'  i2".i±o".3, 

and  that  at  another  culmination  we  find  for  the  same  quantity 
14°  53'  I4".3  ±  o".5. 

In  this  case  we  have  d=  2.2.     This  is  about  3.8  times  the  rela 
tive  probable  error  whose  value  is  o".58. 


82  INDEPENDENT  DETERMINATIONS.          [Art.  102 

From  Table  II  we  find  that  the  probability  that  the  relative 
accidental  error  should  be  as  great  as  d  is  only  about  i  in  100. 
We  are  therefore  justified  in  assuming  that  the  difference  d  is 
mainly  due  to  errors  peculiar  to  the  culminations.  In  other 
words,  we  assume  that,  could  we  have  obtained  the  precise 
values  corresponding  to  the  two  culminations,  (by  indefinitely 
increasing  the  number  of  observations  at  each,)  they  would  still 
be  found  to  differ  by  about  2". 2.  Supposing  now  that  there  is 
no  reason  for  preferring  one  of  these  precise  values  to  the  other, 
we  ought  to  take  their  simple  arithmetical  mean  for  the  final 
result ;  and,  since  the  two  given  values  are  comparatively  close 
to  the  precise  values  in  question,  we  may  take  their  arithmetical 
mean,  which  is 

14°  53'  i3".2, 

for  the  final  determination. 

103.  Determinations  like  those  considered  above,   whose 
difference  is  so  great  as  to  indicate  an  actual  difference  between 
the  precise  values  to  which  they  tend,  may  be  called  discordant 
determinations.     The  discordance  of  the  two  determinations 
discloses  the  existence  of  systematic  errors  which  were  not 
indicated  by  the  discrepancies  of  the  observations  upon  which 
the  given  probable  errors  were  based.   In  combining  the  deter- 
minations, these  systematic  errors   are  treated  as   accidental 
errors  incident  to  the  two  determinations  considered  as  two 
observed  values  of  the  required  quantity.   In  fact,  it  is  generally 
the  object  in  making  new  and  independent  determinations  to 
eliminate  as  far  as  possible  a  new  class  of  errors  by  bringing 
them   into  the  category  of  accidental  errors  which  tend  to 
neutralize  each  other  in  the  final  result.     The  probable  error 
of  the  result  cannot  now  be  derived  from  the  given  probable 
errors,  but  must  be  inferred  from  the  determinations  themselves 
considered  as  observed  values,  because  we  now  take  cognizance 
of  errors  which  are  not  indicated  by  the  given  probable  errors. 

104.  When  there  are  but  two  observed  values,  formula  (4), 
Art.  72,  becomes 


§  VIL]  DISCORDANT  DE  TERMTNA  TIONS.  83 

in  which  pi ,  pz  are  the  weights  assigned  to  the  two  values. 
Denoting  the  difference  by  d,  the  residuals  have  opposite  signs, 
and  their  absolute  values  are 


Substituting  these  values,  we  have  for  the  probable  error  of  the 
mean 

••••  <•> 

When  A  =  A »  this  becomes 

^0=-^=  0.3372* (2) 

In  the  example  given  in  Art.  102,  the  value  of  RQ  thus  obtained 
is  o".742,  which,  owing  to  the  discordance  of  the  two  given 
determinations,  considerably  exceeds  each  of  the  given  probable 
errors. 

Of  course  no  great  confidence  can  be  placed  in  the  results 
given  by  the  formulae  above  on  account  of  the  small  value  of  n? 

105.  Since  the  error  of  each  determination  is  the  sum  of  its 
accidental  and  systematic  error,  if  sl  and  s2  denote  the  probable 

*The  argument  by  which  it  is  shown  that  the  value  of  h  deduced  in 
Art.  69  is  the  most  probable  value  involves  the  assumption  that  before 
the  observations  were  made  all  values  of  h  are  to  be  regarded  as  equally 
probable ;  just  as  that  by  which  it  is  shown  that  the  arithmetical  mean 
is  the  most  probable  value  of  the  observed  quantity  a  involves  the  assump- 
tion that  before  the  observations  all  values  of  a  were  equally  probable.  In 
the  case  of  #,  the  assumption  is  admissible  with  respect  to  all  values  of  a 
which  can  possibly  come  in  question.  But,  in  the  case  of  h,  this  is  not  true  ; 
because  (supposing  n  =  2  as  above)  when  d  =  o  the  value  of  h  is  infinite, 
and  when  d  is  small  the  corresponding  values  of  h  are  very  large,  so  that 
it  is  impossible  to  admit  that  all  values  of  h  which  can  arise  are  h  priori 
equally  probable. 

In  the  present  application  of  the  formula,  however,  these  inadmissible 
values  do  not  arise,  because  we  do  not  use  it  when  d  is  small,  employing 
instead  the  method  of  Art.  99  and  the  formula  of  Art.  xoz. 


84  INDEPENDENT  DETERMINATIONS-          [Art.  105 

systematic  errors,  the  probable  errors  of  the  two  determinations 
when  both  classes  of  errors  are  considered  are 


The  proper  ratio  of  weights  with  which  the  determinations 
should  be  combined  is  R\  :  R\.  The  method  of  procedure 
followed  in  Art.  99  assumes  that  sl  and  $2  vanish.  On  the  other 
hand,  in  the  process  employed  in  Art.  102  we  are  guided,  in  an 
assumption  of  the  ratio  R\  :  R\,  by  a  consideration  of  the  value 
which  the  ratio  s\  :  sl  ought  to  have. 

For  example,  in  the  illustration,  Art.  102,  the  ratio  R\  :  R\  is 
taken  to  be  one  of  equality,  whereas  the  hypothesis  we  desired 
to  make  was  that  Si  =  sz,  so  that  we  ought  to  have 

R\-  Rl  =  r\-  r\. 

On  the  hypothesis  Rl  =  R»  the  value  of  each  of  these  prob- 
able errors  is,  in  accordance  with  equation  (2),  Art.  104,  pd.  In 
the  example  this  is  i".O5.  If  we  take  (i.c»5)2as  the  average 
value  of  Rl  and  R\,  and  introduce  the  condition  written  above, 
we  shall  find  as  a  second  approximation  to  the  value  of  the  ratio 
R\\  Rl  about  15:13.  The  final  value  corresponding  to  this 
ratio  of  weights  is  14°  53'  13".!,  and  its  probable  error  as  deter- 
mined by  equation  (i),  Art.  104,  is  slightly  less  than  that  before 
found,  namely,  R0  =  o".y4O. 

Indicated  and  Concealed  Portions  of  the  Risk  of  Error. 

I06.  It  will  be  convenient  in  the  following  articles  to  speak 
of  the  square  of  the  probable  error  as  the  measure  of  the  risk 
of  error. 

The  foregoing  discussion  shows  that  the  total  risk  of  error, 
R*y  of  any  determination  consists  of  two  parts,  r2  and  s*,  of 
which  the  first  only  is  indicated  by  discrepancies  among  the 
observations  of  which  the  given  determination  is  the  mean.  It 
is  only  this  first  part  that  can  be  diminished  by  increasing  the 
number  of  the  constituent  observations.  The  remaining  part 
remains  concealed,  and  cannot  be  diminished  until  some  varia- 


§  VII.]       PORTIONS'OF  THE   TOTAL  RISK  OF  ERROR.        85 

tion  is  made  in  the  circumstances  under  which  the  observations 
are  made,  giving  rise  to  new  determinations.  When  the  indi- 
cated portions  of  the  risk  of  error  in  the  several  determinations 
are  sufficiently  diminished,  discordance  between  them  must 
always  be  expected,  and  this  discordance  brings  into  evidence 
a  new  portion,  but  still  it  may  be  only  a  portion,  of  the  hitherto 
concealed  part  of  the  risk  of  error. 

107.  What  we  have  called  in  Art.  103  discordant  determina- 
tions are  those  in  which  the  indication  of  this  new  portion  of 
the  risk  of  error,  to  which  corresponds  the  relative  systematic 
error,  is  unmistakable,  because  of  its  magnitude  in  comparison 
with  what  remains  of  the  portion  first  indicated  in  the  separate 
determinations,  that  is,  r\  and  r\.  On  the  other  hand,  the  con- 
cordant determinations  of  Art.  101  are  those  in  which  the  new 
portion  is  so  small  compared  with  r(  and  r\  as  to  remain  con- 
cealed. 

Thus,  to  return  to  the  illustration  discussed  in  Art.  99,  if 
twenty  times  as  many  observations  had  been  involved  in  the 
determination  by  the  transit,  its  probable  error  would  have 
been  reduced  to  equality  with  that  of  the  determination  by  the 
theodolite.  But  if  this  had  been  done  we  should  almost  cer- 
tainly have  found  the  determinations  discordant ;  that  is  to  say, 
the  ratio  in  which  the  difference  between  the  determinations  is 
reduced  would  be  much  less  than  that  in  which  the  probable 
relative  accidental  error  \j  (r\  +  r[)  is  diminished.  The  ratio  in 
which  the  remaining  difference  between  the  determinations 
should  be  divided  in  making  the  final  determination  now 
depends  upon  our  estimate  of  the  comparative  freedom  of  the 
instruments  from  systematic  error,*  but  the  important  thing  to 
be  noted  is  that  the  probable  error  of  the  result  would  now  be 
found  as  in  Art.  104,  and  would  be  greater  than  those  of  the 

*It  may  be  assumed  that,  when  the  instruments  are  carefully  adjusted, 
the  one  which  is  less  liable  to  accidental  errors  is  correspondingly  less 
liable  to  systematic  errors.  But  this  comparison  is  concerned  with  the 
probable  errors  of  a  single  observation  in  each  case,  and  not  with  those  of 
the  determinations  themselves. 


86  INDEPENDENT  DETERMINATIONS.         [Art.  107 

separate  determinations.  Thus  the  apparent  risk  of  error  would 
be  increased  by  making  a  new  determination,  but  this  is  only 
because  a  greater  part  of  the  total  risk  of  error  has  been  made 
apparent,  and  the  result  is  so  much  the  more  trustworthy  as  a 
greater  variety  has  been  introduced  into  the  methods  employed. 

The  Total  Probable  Error  of  a  Determination. 

I08.  In  the  illustrations  given  in  Arts.  99  and  102  it  was  sup- 
posed that  two  determinations  only  were  made,  so  that  we  had 
but  a  single  discrepancy  upon  which  to  base  our  judgment  of  the 
probable  amount  of  the  relative  systematic  error.  But,  in  general, 
what  are  regarded  as  determinations  at  one  stage  of  the  process 
are  at  the  next  stage  treated  as  observations  which  may  be 
repeated  indefinitely  before  being  combined  into  a  new  deter- 
mination. Let  one  of  the  determinations  first  made  be  the 
mean  of  n  observations  equally  good,  and  let  r  be  the  probable 
error  of  a  single  observation.  Then  the  probable  accidental 

error  of  the  mean  is  r0  =  -£-  .     Now,  if  R  is  the  probable  error 

of  the  final  value  as  obtained  directly  from  the  discrepancies 
of  the  several  determinations,  (their  number  being  supposed 
great  enough  to  allow  us  to  obtain  a  trustworthy  value,)  we  shall 
find  that  R  exceeds  r0,  and  putting 


r\  is  the  new  portion  of  the  risk  of  error  brought  out  by  the 
comparison  of  the  determinations. 

*^2 

IOp.  The  form  of  this  equation  shows  that  when  —  is  already 

small  compared  with  r\,  the  advantage  gained  by  increasing  the 
value  of  n  soon  becomes  inappreciable. 

For  example,  the  reticule  of  a  meridian  circle  is  provided 
with  a  number  of  threads,  in  order  that  several  observations  of 
time  may  be  taken  at  a  single  transit.  If  seven  equidistant  threads 
are  used,  the  mean  of  the  times  is  equivalent  to  a  determination 


§  VII.]  THE   TOTAL  PROBABLE  ERROR.  87 

based  upon  seven  observations  of  the  time  of  transit.  Chauvenet 
found  that,  for  moderately  skilful  observers,  the  probable  acci- 
dental error  of  the  transit  over  a  single  thread  of  an  equatorial 
star  is  r  =  os.o8,  whence  for  the  mean  of  the  seven  threads  we 
have  r0  =  os.c»3.  The  probable  error  of  a  single  determination 
of  the  right  ascension  of  an  equatorial  star  was  found  to  be 
R  =  o'.o6,  so  that,  from  Rz  =  rl  +  r\  we  have  r^  =  os.o52.  The 
conclusion  is  reached  that  "  an  increase  of  the  number  of  threads 
would  be  attended  by  no  important  advantage,"  and  it  is  stated 
that  Bessel  thought  five  threads  sufficient* 

110.  Suppose  the  value  of  R*  in  equation  (i),  Art.  108,  to 
have  been  derived  from  the  discrepancies  of  nr  determinations  of 
equal  weight.  A  systematic  error  may  exist  for  these  n' 
determinations,  and  sl  being  its  probable  value,  we  shall  have 


that  is  to  say,  the  concealed  portion  of  the  risk  of  error  in  one 
of  the  original  determinations  has  been  decomposed  into  two 
parts,  one  of  which  has  been  disclosed  at  the  second  stage  of  the 
process,  while  the  other  remains  concealed. 

The  total  risk  of  error  in  a  single  one  of  the  ri  determina- 
tions is  R*  +  s\t  and  that  of  the  mean  of  the  determinations  is 

§*«. 

In  like  manner,  if  at  a  further  stage  of  the  process  we  have  the 
means  of  finding  the  value  of  the  probable  error  JKi  of  this  new 
determination  by  direct  comparison  with  other  coordinate  deter- 
minations, a  portion  of  the  value  of  s\  will  be  disclosed,  and  we 
shall  have 


where  again  it  must  be  supposed  that  a  portion  si  of  the  risk  of 
error  still  remains  concealed. 

*  Chauvenet's  "  Spherical   and  Practical   Astronomy,"  vol.  ii,  p.  194 
et  seq. 


88  INDEPENDENT  DE  TERMINA  TIONS.  [Art.  1 1 1 

111.  The  comparative  amounts  of  the  risk  of  error  which  are 
disclosed  at  the  various  stages  of  the  process  depend  upon  the 
amount  of  variety  introduced  into  the   method  of  observing. 
Thus,  to  resume  the  illustration  given  in  Art.  109,  if  the  star 
be  observed  at  n'  culminations,  r*  will  correspond  to  errors 
peculiar  to  a  thread,  and  r\  will  correspond  to  errors  peculiar  to  a 
culmination.     Again,  if  different  stars  whose  right  ascensions  are 
known  are  observed,  in  order  to  obtain  the  local  sidereal  time 
used  in  a  determination  of  the  longitude,  v\  will  correspond  to 
errors  peculiar  to  a  star,  together  with  instrumental  errors 
peculiar  to  the  meridian  altitude. 

The  Ultimate  Limit  of  Accuracy. 

112.  The  considerations  adduced  in  the  preceding  articles 
seem  to  point  to  the  conclusion  that  there  must  always  be  a 
residuum  of  the  risk  of  error  that  has  not  yet  been  reached,  and 
thus  to  explain  the  apparent  existence  "  of  an  ultimate  limit  of 
accuracy  beyond  which  no  mass  of  accumulated  observations 
can  ever  penetrate."*     But  it  does  not  appear  to  be  necessary 
to  suppose,  as  done  by  Professor  Peirce,  that  there  is  an  absolute 
fixed  limit  of  accuracy,  due  to  "  a  failure  of  the  law  of  error 
embodied  in  the  method  of  Least  Squares,  when  it  is  extended 
to  minute  errors."     He  says:   "In  approaching  the  ultimate 
limit  of  accuracy,  the  probable  error  ceases  to  diminish  propor- 
tionally to  the  increase  of  the  number  of  observations,  so  that 
the  accuracy  of  the  mean  of  several  determinations  does  not 
surpass  that  of  the  single  determinations  as  much  as  it  should 
do,  in  conformity  with  the  law  of  least  squares ;  thus  it  appears 
that  the  probable  error  of  the  mean  of  the  determinations  of  the 
longitude  of  the  Harvard  Observatory,  deduced  from  the  moon- 
culminating  observations  of  1845,  1846,  and  1847,  is  is.28  instead 
of  1 8.oo,  to  which  it  should  have  been  reduced  conformably  to 
the  accuracy  of  the  separate  determinations  of  those  years." 

*  Prof.  Benjamin  Peirce,  U.  S.  Coast  Survey  Report  for  1854,  Appendix, 
p.  109, 


§  VII.]  EXAMPLES.  89 

To  account  for  the  fact  cited  on  the  principles  laid  down 
above,  it  is  only  necessary  to  suppose  that  there  are  causes  of 
error  which  have  varied  from  year  to  year ;  and,  recognizing  this 
fact,  we  ought  to  obtain  our  final  determination  by  comparing 
the  determinations  of  a  number  of  years,  and  not  by  combining 
into  one  result  the  whole  mass  of  observations. 

Examples. 

1.  In  a  system  of  observations  equally  good,  r  being  the 
probable  error  of  a  single  observation,  if  two  observations  are 
selected  at  random,  what  quantity  is  their  difference  as  likely  as 
not  to  exceed  ?  r  tf  2. 

2.  In  example  i,  what  is  the  probability  that  the  difference 
shall  be  less  than  r?  0.367. 

3.  When  two  determinations  are  made  by  the  same  method, 
show  that  the  odds  are  in  favor  of  a  difference  less  than  the  sum 
of  the  two  probable  errors,  and  against  a  difference  less  than  the 
greater  of  the  two,  and  find  the  extreme  values  of  these  odds. 

66 :  34  and  63 : 37. 

4.  A  and  B  observe  the  same  angle  repeatedly  with  the  same 
instrument,  with  the  following  results : 

A  B 

47°  23'  40"  47°  23'  30" 

47    23  45  47    23  40 

47    23  30  47    23  50 

47    23  35  47    24  oo 

47    23  40  47    23  20 

Show  that  there  is  no  evidence  of  relative  systematic  (personal) 
error.  Find  the  relative  weights  of  an  observation  by  A  and 
by  B,  and  the  final  determination  of  the  angle. 

100 : 13;  47°  23'  38".23  ±  i".62. 

5.  Show  that  the  probable  error  in  example  4  as  computed 
from  the  ten  observations  taken  with  their  proper  weights  is 
i"-53,  but  that  derived  from  the  formula  of  Art.  104  is  0^.43, 
which  is  much  too  small.     (See  foot-note,  p.  83.) 


90  INDEPENDENT  DE  TERMINA  TIONS.         [Art.  1 12 

6.  Two  determinations  of  the  length  of  a  line  in  feet  give 
respectively  683.4  ±0.3  and  684.9  ±  0.3,  there  being  no  reason 
for  preferring  one  of  the  corresponding  precise  values  to  the 
other ;  show  that  the  probable  error  of  each  of  the  precise  values 
(that  is,  the  systematic  error  of  each  determination)  is  0.65 ;  and 
that  the  best  final  determination  is  684.15  ±0.51. 

7.  Show  generally  that  when  the  weights  are  inversely  pro- 
portional to  the  squares  of  the  probable  errors,  the  formula  of 
Art.  104  gives  a  value  of  R  greater  or  less  than  that  given  by 
the  formula  of  Art.  101,  according  as  d  is  greater  or  less  than 
the  relative  mean  error. 


VIII. 

INDIRECT  OBSERVATIONS. 
Observation  Equations. 

113.  We  have    considered    the    case  in    which  a  quantity 
whose  value  is  to  be  determined  is  directly  observed,  or  is 
expressed  as  a  function  of  quantities  directly  observed.    We 
come  now  to  that  in  which  the  quantity  sought  is  one  of  a 
number  of  unknown  quantities  of  which  those  directly  observed 
are  functions.     The  equation  expressing  that  a  known  function 
of  several  unknown  quantities  has  a  certain  observed  value  is 
called  an  observation  equation.     Let  i*.  denote  the  number  of 
unknown  quantities  concerned.     Then,  in  order  to  determine 
them,  we  must  have  at  least  /*  independent  equations.    Thus, 
if  two  of  the  equations  express  observed  values  of  the  same 
function  of  the  unknown  quantities,  they  will  either  be  ident- 
ical, so  that  we  have  in  effect  only  p. —  i  equations,  or  else  they 
will  be  inconsistent,  so  that  the  values  of  the  unknown  quan- 
tities will  be  impossible.     So  also  it  must  not  be  possible  to 
derive  any  one  of  the  ^  equations,  or  one  differing  from  it  only 
in  the  absolute  term,  from  two  or  more  of  the  other  equations. 

114.  If  we  have  no  more  than  the  necessary  i*.  equations,  we 
shall  have  no  indication  of  the  precision  with  which  the  obser- 
vations have  been  made,  nor,  consequently,  any  measure  of  the 
precision  with  which  the  unknown  quantities  have  been  deter- 
mined.   With  respect  to  them,  we  are  in  the  same  condition  as 
when  a  single  observed  value  is  given  in  the  case  of  direct 
observations. 

Now  let  other  observation  equations  be  given,  that  is  to  say, 
let  the  values  of  other  functions*  of  the  unknown  quantities  be 
observed.  The  results  of  substituting  the  values  of  the  unknown 

*  It  is  not  necessary  that  these  additional  equations  should  be  inde- 
pendent of  the  original  /z  equations,  for  an  equation  expressing  a  new 
observed  value  of  a  function  already  observed  will  be  useful  in  deter- 
mining the  precision  of  the  observations. 


92  INDIRECT  OBSERVATIONS.  [Art.  114 

quantities  will,  owing  to  the  errors  of  observation,  be  found  tc 
differ  from  the  observed  values,  and  the  discrepancies  will  give 
an  indication  of  the  precision  of  the  observations,  just  as  the  dis- 
crepancies between  observed  values  of  the  same  quantity  do,  :n 
the  case  of  direct  observations. 

115.  As  an  example,  let  us  take  the  following  four  observa- 
tion equations*  involving  x,  y  and  z  : 

x—  y  +  22  =    3, 

3-r  +  zy  -  52  =    5, 

4_r  +  y  4-  42=  21, 

~  x  +  37  +  3*  =  14. 

if  we  solve  the  first  three  equations  we  shall  find 


Substituting  these  values  in  the  fourth  equation,  the  value  of 
the  first  member  is  I2f,  whereas  the  observed  value  is  14;  the 
discrepancy  is  i^.  If  the  values  above  were  the  true  values, 
the  errors  of  observation  committed  must  have  been  o,  o,  o,  i^; 
but,  since  each  of  the  observed  quantities  is  liable  to  error,  this 
is  not  a  likely  system  of  errors  to  have  been  committed.  In 
fact,  any  system  of  values  we  may  assign  to  xty  and  z  implies 
a  system  of  errors  in  the  observed  quantities,  and  the  most 
probable  system  of  values  is  that  to  which  corresponds  the 
most  probable  system  of  errors. 

Il6.  In  general,  let  there  be  m  observation  equations, 
involving  p  unknown  quantities,  m>f*',  then  we  have  first  to 
consider  the  mode  of  deriving  from  them  the  most  probable 
values  of  the  unknown  quantities.  The  system  of  errors  in  the 
observed  quantities  which  this  system  of  values  implies  will 
then  enable  us  to  measure  the  precision  of  the  observations. 
Finally.,  regarding  the  n  unknown  quantities  as  functions  of  the 
m  observed  quantities,  we  shall  obtain  for  each  unknown  quan- 
tity a  measure  of  the  precision  with  which  it  has  been  deter- 
mined. 

*  Gauss,  "  Theoria  Motus  Corporum  Coelestium,"  Art.  184. 


§VIIL]  REDUCTION1  TO  LINEAR  FORM.  93 

The  Reduction  of  Observation  Equations  to  the  Linear  Form. 

117-  The  method  of  obtaining  the  values  of  the  unknown 
quantities,  to  which  we  proceed,  requires  that  the  observation 
equations  should  be  linear.  When  this  is  not  the  case,  it  is 
necessary  to  employ  approximately  equivalent  linear  equations, 
which  are  obtained  in  the  following  manner. 

Let  X,  Y,  Z,  .  .  ,  be  the  unknown  quantities,  and  Mt, 
J/2,  . . .  Mm  the  observed  quantities;  the  observation  equations 
are  thec,  of  the  form 


where  /"?,/"2,  .  .  .fm  are  known  functions.  Let  X0,  Y0,  Z0,  .  .  . 
be  approximate  values  of  X,  Y,  Z,  .  .  .,  which,  if  not  otherwise 
known,  may  be  found  by  solving  P.  of  the  equations;  and  put 

X=X0  +  x,         Y=  Y0+y,     ..., 

so  that  x?yyz,...  are  small  corrections  to  be  applied  to  the 
approximate  values.  Then  the  first  observation  equation  may 
be  written 


x, 
or,  expanding  by  Taylor's  theorem, 


where  the  coefficients  of  x,  y,  z  ,  .  .  .  are  the  values  which  the 
partial  derivatives  of  fi(X,  Y,  Z,  .  .  .  )  assume  when  X—  X0, 
Y—Y0yZ—Z0,.a,,  and  the  powers  and  products  of  the 
small  quantities  x,y,  z,  .  .  .  are  neglected  as  in  Art.  91. 

Denoting  the  coefficients  of  x,  y,  2,  *  .  ,  by  «i,  £t  ,  <i,  .  .  .  , 
putting  ^i  for  M^  —  /i  (X0  ,  K>  ,  Z0  ,  .  ,  „  ),  and  treating  the  other 
observation  equations  in  the  same  way,  we  may  write 


94  INDIRECT  OBSERVATIONS.  [Art    117 


=HI   "1 


.  „  .         2 

.  •  •   =  nm  J 


for  the  observation  equations  in  their  linear  form« 

Il8.  Even  when  the  original  observation  equations  are  in  th^ 
linear  form,  it  is  generally  best  to  transform  them  as  aboves  su 
that  the  values  of  the  unknown  quantities  shall  be  small. 

Another  transformation  sometimes  made  consists  in  replacing 
one  of  the  unknown  quantities  by  a  fixed  multiple  of  it.  For 
example,  if  the  values  of  the  coefficients  oiy  are  inconveniently 
large  they  may  be  reduced  in  value  by  substituting  ktf  fory 
and  giving  to  k  a  suitably  small  value. 

Up.  In  the  observation  equations  (i),  the  second  members 
may  be  regarded  as  the  observed  quantities,  since  they  have  the 
same  errors.  If  the  true  values  of  x>y>  z  ?  o  „  .  are  substituted  in 
these  equations  they  will  not  be  satisfied,  because  each  n  differs 
from  its  proper  value  by  the  error  of  observation  v  \  we  may 
therefore  write  the  equations 


c-js 


>,.       •       .       (2) 


in  which,  \ix,y,  z,  .  .  .  are  the  true  values,  v^ ,  fy, .  .  „  vm  are  the 
true  errors  of  observation,  and  if  any  set  of  values  be  given  to 
Xj  y,  z,  .  .  . ,  the  second  members  are  the  corresponding  resid- 
uals. These  corrected  observation  equations  may  be  called  the 
residual  equations. 

Observation  Equations  of  Equal  Precision. 

120.  Let  us  first  suppose  that  the  m  observations  are  equally 
good,  and  let  h  be  their  common  measure  of  precision.  Then, 
since  v  ir  the  error,  not  only  of  the  absolute  term  n±  in  the  first 
of  equations  (2),  but  of  the  first  observed  quantity  Mlt  the  prob- 


§VIIL]  EQUATIONS  OF  EQUAL  PRECISION.  95 

ability  before  the  observations  are  made  that  the  first  observed 
value  shall  be  Ml  is 


where,  as  in  Art.  35,  Av  is  the  least  count  of  the  instrument. 
Hence  we  have,  for  the  probability  before  the  observations  are 
made  that  the  m  actual  observed  values  shall  occur, 


exactly  as  in  Art.  41.  The  values  of  v\  ,  z£  ,  .  .  .  v^  being  given 
by  equations  (2),  this  value  of  -Pis  a  function  of  the  several 
unknown  quantities  ;  hence  it  follows,  as  in  Art.  41,  that  for  any 
one  of  them  that  value  is,  after  the  observations  have  been 
made,  most  probable  which  assigns  to  P  its  maximum  value  ; 
in  other  words,  that  value  which  makes 

^i+^+  •••  +^  =  a  minimum. 

Thus  the  principle  of  Least  Squares  applies  to  indirect  as 
well  as  to  direct  observations. 

121.  To  determine  the  most  probable  value  of  x,  we  have,  by 
differentiation  with  respect  to  x, 

dv,  dv«  dvm 

*'S+*s  +  ---  +  fS&as0' 

or,  since,  from  equations  (2),  Art.  119, 

dvl  dvz  dvn 


.  0  .  +  amvm  =  o  .....     (i) 

This  is  called  the  normal  equation  for  x.  Whatever  values 
are  assigned  to  y,  z,  .  .  .  ,  it  gives  the  rule  for  determining  the 
value  of  x  which  is  most  probable  on  the  hypothesis  that  the 
values  assigned  to  the  other  unknown  quantities  are  correct. 

Since  vltvt,...vm  represent  the  first  members  of  the  obser- 


g6  INDIKE C  T  OBSER  VA  TIONS.  [ Art.  1 2 1 

vation  equations  (i),  Art.  117,  when  so  written  that  the  second 
member  is  zero,  we  see  that  the  normal  equation  for  x  may  be 
formed  by  multiplying  each  observation  equation  by  the  coeffi- 
cient of  x  in  it,  and  adding  the  results. 

122.  The  rule  just  given  for  forming  the  normal  equation 
shows  it  to  be  a  linear  combination  of  the  observation  equations, 
and  the  reason  why  the  multipliers  should  be  as  stated  may  be 
further  explained  as  follows:  If  we  suppose  fixed  values  given 
to  j/,  z, .  .  .  ,  each  observation  equation  may  be  written  in  the 
form  ax  =  Ny  where  N  only  differs  from  the  observed  value 
M  by  a  fixed  quantity,  and  therefore  has  the  same  probable 
error.  Now,  writing  the  observation  equations  in  the  form 


we  may  regard  them  as  expressing  direct  observations  of  x.  If 
r  is  the  common  probable  error  of  A£ ,  Nz,  .  .  .  Nmt  that  of 

— -  or  x^  is  — ;  that  of  x^  is  — ,  and  so  on.     Thus  the  equations 

«!  «!  fl3 

are  not  of  equal  precision  for  determining  x,  and  their  weights 
when  written  as  above  (being  inversely  as  the  squares  of  the 
probable  errors)  are  as  a\ :  a\ :  .  .  .  :  a*m.  It  follows  that  the 
equation  for  finding  x  is,  as  in  the  case  of  the  weighted  arith- 
metical mean  (see  Art.  66),  the  result  of  adding  the  above 
equations  multiplied  respectively  by  a?,  d£,  ..,*£;*  that  is  to 
say,  it  is  the  result  of  adding  the  original  observation  equations 
of  the  form  ax  —  N—o>  multiplied  respectively  by  alt  azy .  . .  am. 

*It  must  not  be  assumed  that  the  weight  of  the  value  of  x,  determined 
from  the  several  normal  equations,  is  202,  that  of  an  observation  being 
unity.  This  is  its  weight  only  upon  the  supposition  that  the  absolute 
values  of  the  other  quantities  are  known. 


§VIII.]  THE  NORMAL  EQUATIONS.  97 

The  Normal  Equations. 

123.  In  like  manner,  for  each  of  the  other  unknown  quantities 
we  can  form  a  normal  equation,  and  we  thus  have  a  system  of 
equations  whose  number  is  equal  to  that  of  the  unknown  quan- 
tities. The  solution  of  this  system  of  normal  equations  gives 
the  most  probable  values  of  the  unknown  quantities.  Let  us 
take  for  example  the  four  observation  equations  given  in  Art. 
115.  Forming  the  normal  equations  by  the  rule  given  above, 
we  have 

27^  +    6jj/  =    88, 

6x  +  i5jv  +      z  =    70, 

y  +  54^  =  107. 

The  solution  of  this  system  of  equations  gives  for  the  most 
probable  values, 


z—  -t~-    —  1.92. 
6633 

124.  Writing  the  observation  equations  in  their  general  form, 


...  -f  IJ  =  nv 
.  .  .  +  /„/  =  n.2 


+  .  .  .  +  m—  n 
we  obtain  for  the  normal  equations  in  their  general  form, 

Za?  .x+  lab.y  +  .  .  .  +  lal.t-  Ian   1 

Iab.x+  IP   .y  +  .  .  .  4-  161.  t  =  Ibn    \  _     B     ^ 

lal  .  x+  Ibl  .y  +  .  .  .  +  IP  .  /  =  Hn    \ 

It  will  be  noticed  that  the  coefficient  of  the  rth  unknown 
quantity  in  the  .yth  equation  is  the  same  as  that  of  the  ^th 
unknown  quantity  in  the  rth  equation;  in  other  words,  the 


98  INDIRECT  OBSERVATIONS.  [Art.  124 

determinant  of  the  coefficients  of  the  unknown  quantities  in 
equations  (2)  is  a  symmetrical  one. 

Observation  Equations  of  Unequal  Precision. 

125.  When  the  observations  are  not  equally  good,  if 

^i,  &2,  .  .  •  .  km 
are  the  measures  of  precision  of  the  observed  values 

M^  M2,  .  .  .  Mm, 
the  expression  to  be  made  a  minimum  is 

h\v\  +  h\v\  +  .  .  .  +  Ai«i, 

as  in  Art.  65.  Thus,  as  in  the  case  of  direct  observations,  if  the 
error  of  each  observation  be  multiplied  by  its  measure  of  pre- 
cision so  as  to  reduce  the  errors  to  the  same  relative  value,  it  is 
necessary  that  the  sum  of  the  squares  of  the  reduced  errors 
should  be  a  minimum. 

Since  z/j  =  o,  vz  =  o,  .  .  .  vm  =  o  are  equivalent  to  the  observa- 
tion equations,  it  follows  that,  if  we  multiply  each  observation 
equation  by  its  measure  of  precision  (so  that  it  takes  the  form 
hv  =  o),  we  may  regard  the  results  as  equations  of  equal  pre- 
cision. 

126.  The  result  may  be  otherwise  expressed  by  using  num- 
bers /!  ,  pz  ,  .  .  .  pm  proportional,  as  in  Art.  66,  to  the  squares  of 
the  measures  of  precision  ;  the  quantity  to  be  made  a  minimum 
then  is 


and  the  normal  equation  for  x  is 

p\a\v\  +  p*a&i  +  .  .  .  +  pmamvm  =  o. 

The  numbers  p±  ,  p2  ,  .  .  .  pm  are  called  the  weights  of  the 
observation  equations;  thus,  in  the  case  of  weighted  equations, 
the  normal  equation  for  x  may  be  formed  by  multiplying  each 
observation  equation  by  the  coefficient  of  x  in  it,  and  also  by  its 
weight,  and  adding  the  results. 


§VIIL]        EQUATIONS  OF  UNEQUAL  PRECISION.  99 

The  general  form  of  the  normal  equations  is  now 

Ipa*  .x  +  Ipab.y  +  .  .  .  +  Ipal.t—Ipan 

Ipab  .  x  +  ipb*  .y  +  .  .  .  +  Ipbl  .  t  =  Ipbn 

Ipal  .  x  +  Ipbl  .>+...  +  I>/2  .  /  =  Ipln 

The  result  is  evidently  the  same  as  if  each  observation  equation 
had  been  first  multiplied  by  the  square  root  of  its  weight,  by 
which  means  it  would  be  reduced  to  the  weight  unity,  and  the 
system  would  take  the  form  (2),  Art.  124. 

Formation  of  the  Normal  Equations. 

127.  When  the  normal  equations  are  calculated  by  means  of 
their  general  form,  a  table  of  squares  is  useful  not  only  in  cal- 
culating the  coefficients  Ipa1,  Ipb*  ,  .  .  .  IpP,  but  also  in  the 
case  of  those  of  the  form  Ipab,  Ipac,  .  .  .  Ipan,  .  .  .  For, 
since 

ab=  *[(«  +  J)1-**-  Ja], 
we  have 

Ipab  =  ^Ip(a  +  by  -  Iptf  -  Ipb^, 

by  means  of  which  Ipab  is  expressed  in  terms  of  squares.*  Or 
for  the  same  purpose  we  may  use 

Ipab  =  $\_Ipa*  +  Ipb*  -  Ip(a  -  3)2]. 

In  performing  the  work  it  is  convenient  to  arrange  the  coeffi- 
cients in  a  tabular  form  in  the  order  in  which  they  occur  in  the 
observation  equations,  and,  adding  a  column  containing  the  sums 
of  the  coefficients  in  each  equation,  thus, 

*i  =  «i  +  £i  +  ...  +/i  +  wn    etc., 
*  If  'Zpab  alone  were  to  be  found,  the  formula 


derived  from  that  of  quarter-squares,  would  be  preferable  ;  but,  since 
2/>a2,  2/£*  have  also  to  be  calculated,  the  use  of  the  formula  above, 
which  was  suggested  by  Bessel,  involves  less  additional  labor. 


100 


INDIRECT  OBSERl'A  1  IO\S. 


[Art.  12; 


to  form  the  quantities  *2pas,  *2pbs, .  .  .  ^pns  in  addition  to  those 
which  occur  in  the  normal  equations.     We  ought  then  to  find 

Ipas  =  Ipc?  +  Ipab  +  .  .  .  +  Ipan, 
Ipbs  -  Zpab  +  Zpb*  +  .  .  .  +  Ipbn, 


-  .  .  .  +  lpn\ 

and  the  fulfilment  of  these  conditions  is  a  verification  of  the 
accuracy  of  the  work. 

In  many  cases,  the  use  of  logarithms  is  to  be  preferred, 
especially  when  the  logarithms  of  the  coefficients  in  the  ob- 
servation equations  are  more  readily  obtained  than  the  values 
themselves. 

The  General  Expressions  for  the  Unknoum  Quantities. 

128.  In  writing  general  expressions  for  the  most  probable 
values  of  the  unknown  quantities,  and  in  deriving  their  prob- 
able errors,  we  shall,  for  simplicity  in  notation,  suppose  that  the 
observation  equations  have  been  reduced  to  the  weight  unity  as 
explained  in  Art.  126,  so  that  they  are  represented  by  equations 
(i),  and  the  normal  equations  by  equations  (2)  of  Art.  124. 

Let  D  be  the  symmetrical  determinant  of  the  coefficients  of 
the  unknown  quantities  in  the  normal  equations,  thus 

la*     lab    ...     lal 
lab    IP  Ibl 


lal    Ibl     ...     II 

let  Dx  denote  the  result  of  replacing  the  first  column  by  a 
column  consisting  of  the  second  members,  Ian,  Ibn,  .  .  .  Iln\ 
and  let  Dy,Dti...  Dt.  be  the  like  results  for  the  remaining 
columns.  Then 


x  = 


D 


t  = 


D  '       •'"  D  '  ~  D 

are  the  general  expressions  for  the  unknown  quantities. 


(i) 


§  VIIL]         EXPRESSIONS  IN  DE  TERM2NANT  FORM.          \ O I 

I2Q.  Let   the  value  of  x  when   expanded   in  terms  of  the 
second  members  of  the  normal  equations  be 


x  =  Qilan  +  Qilbn  +  .  .  .  +  Q^Iln. 


(2) 


Now,  in  the  expansion  of  the  determinant  Dx  in  terms  of  the 
elements  of  its  first  column,  the  coefficients  of  Ian,  Ibn, . . .  2 In 
are  the  first  minors  corresponding  to  Ia?y  lab,  .  .  .  lal,  in  the 
determinant  D. 

Denoting  the  first  of  these  by  D± ,  so  that 

IP     Ibc    ...     Ibl 
Ibc     I?  Id 


Ibl    Id    .. 


it  follows,  on  comparing  the  values  of  x  in  equations  (i)  and 
(2),  that 

&  -  27 ' 

In  like  manner,  the  values  of  £?2,  Q3,  .  .  .  Q^  are  the  results  of 
dividing  the  other  first  minors  by  D. 

The  Weights  of  the  Unknown  Quantities. 

130.  Let  the  value  of  x,  when  fully  expanded  in  terms  of  the 
second  members  nlt  n2,  .  .  .  nm  of  the  observation  equations,  be 


X  = 


+  .  .  .   +  amnm. 


(3) 


Then,  if  rx  denotes  the  probable  error  of  x,  and  r  that  of  a 
standard  observation,  that  is,  the  common  probable  error  of 
each  of  the  observed  values  nlt  n^  .  .  .  nm,  we  shall  have,  by 
Art.  89, 


The  precision  with  which  x  has  been  determined  is  usually 
expressed  by  means  of  its  weight,  that  of  a  standard  observation 


IO2  INDIRECT  OBSERVATIONS.  [Art.  130 

being  taken  as  unity.  The  weights  being  inversely  propor- 
tional to  the  squares  of  the  probable  errors,  we  have,  therefore, 
for  that  of  xt 


131.  Since  the  value  of  x  is  obtained  from  the  normal  equa- 
tions, we  do  not  actually  find  the  values  of  the  a's  ;  we  therefore 
proceed  to  express  JtV  in  terms  of  the  quantities  which  occur 
in  the  normal  equations. 

Equating  the  coefficients  of  HI  ,  n2  ,  .  .  .  nm  in  equations  (2) 
and  (3),  we  find 


.    .    (i) 


Multiplying  the  first  of  these  equations  by  at  ,  the  second  by 
a2  ,  and  so  on,  and  adding  the  results,  we  have 

IV  =  laa  .Ql  +  Zba.Qz+...+  yia.QtL.     .     (2) 

The  value  of  2aa  is  found  by  multiplying  the  first  of  equa- 
tions (i)  by  «!  ,  the  second  by  «2,  and  so  on,  and  adding.  The 
result  is 

laa.  =  Ic?  .  Ql  +  lab.  Qz  +  .  .  .  +  Sal.  Q^     .     (3) 

Multiplying  this  equation  by  D,  the  second  member  becomes 
the  expansion  of  the  determinant  D  in  terms  of  the  elements  of 
its  first  column.  Hence 

2aa  =  i  .........    (4) 

In  like  manner  we  find 


Iba  =  lab.Q,  +  Ib\  Q,+  ...  +  Ibl.  £>M,       .     (5) 

and  when  this  equation  is  multiplied  by  D,  the  second  member 
is  the  expansion  of  a  determinant  in  which  the  first  two  columns 


§VIII.]     WEIGHTS  OF  THE  UNKNOWN  QUANTITIES.      IO3 

are  identical.     Thus  2ba  =  o,  and  in  the  same  way  we  can  show 
that  2ca  ,  .  .  .  2  la  vanish.* 

Substituting  in  equation  (2),  we  have  now 

2a*=&;      .......    (6) 

hence  from  Arts.  130  and  129  we  have,  for  the  general  expres- 
sion for  the  weight  of  x, 


132.  It  follows  from  equation  (2),  Art.  129,  that  if  in  solving 
the  normal  equations  we  retain  the  second  members  in  alge- 
braic form,  putting  for  them  A,  B,  C,  .  .  .  ,  then  the  weight  of  x 
will  be  the  reciprocal  of  the  coefficient  of  A  in  the  value  of  x."\ 
In  like  manner,  that  of  y  will  be  the  reciprocal  of  the  coefficient 
of  B  in  the  value  of  y,  and  so  on. 

For  example,  if  the  normal  equations  given  in  Art.  123  are 
written  in  the  form 


6y  =  A, 

6x  +  i5jy  +      z  =  B, 
y  +  MZ  =  C, 
the  solution  is 

19899*  =    809^  —  324^  -f-      6C, 


66332  =        2  A  —      gB  +  i23'C 


*Comparing  equation  (3)  with  equation  (2),  Art.  129,  we  see  that  2oa 
is  the  value  which  x  would  assume  if  in  each  normal  equation  the 
second  member  were  equal  to  the  coefficient  of  x.  The  system  of  equa- 
tions so  formed  would  evidently  be  satisfied  by  x  =:  \,y  —  o,  z  =  o,  .  .  . 
/  =  o  ;  hence  2aa  =  i.  In  like  manner,  comparing  equation  (5)  with  the 
same  equation,  we  see  that  2<5a  is  the  value  which  x  would  assume  if 
the  second  member  of  each  normal  equation  were  equal  to  the  coefficient 
of  y.  This  value  would  be  zero  ;  thus  2&z  —  o. 

f  If  the  value  of  the  weight  of  x  alone  is  required,  it  may  be  found  as 
the  reciprocal  of  what  the  value  of  x  becomes  when  A  =  i,  B  =  o, 
C  =  o,  .  .  .  ,  that  is  to  say,  when  'the  second  member  of  the  first  normal 
equation  is  replaced  by  unity,  and  that  of  each  of  the  others  by  zero. 


104  INDIRECT  OBSERVATIONS.  [Art.  132 

The  weights  of  x,y  and  z  are  therefore 


=13-65, 
=53-93- 


133.  When  the  value  of  ^r  is  obtained  by  the  method  of  sub- 
stitution, the  process  may  be  so  arranged  that  its  weight  shall  be 
found  at  the  same  time.  Let  the  other  unknown  quantities  be 
eliminated  successively  by  means  of  the  other  normal  equations, 
the  value  of  x  being  obtained  from  the  first  normal  equation  or 
normal  equation  for  x.  Then,  if  this  equation  has  not  been 
reduced  by  multiplication  or  division,  the  coefficient  of  A  in 
the  second  member  will  still  be  unity,  and  the  equation  will  be 
of  the  form 

Rx  =  T  +  A9 

where  T  depends  upon  the  quantities  B,  C,  .  .  .  Now  it  is 
shown  in.  the  preceding  article  that  the  weight  of  x  is  the  recip- 
rocal of  the  coefficient  of  A  in  the  value  of  x  ;  hence  in  the 
present  form  of  the  equation  the  weight  is  the  coefficient  of  jr.* 
As  an  illustration,  let  us  find  the  values  of  x  and  its  weight 
in  the  example  given  above,  the  normal  equation  being 

27*  +    6y  =    88, 

6x  +  i5jj/  +      z  =    70, 

y  +  54*  =  107. 

The  last  equation  gives 

i  107 

z-=  ---  y  H  --  •*•  , 
54^        54 

*The  effect  of  the  substitution  is  always  to  diminish  the  coefficient  of 
x;  for,  as  mentioned  in  the  foot-note  to  Art.  122,  if  the  true  values  of 
y,  0,  .  .  .  /  were  known,  the  weight  of  x  would  be  2a2,  which  is  the  original 
coefficient  of  x,  and  obviously  the  weight  on  this  hypothesis  would  exceed 
px  ,  which  is  the  weight  when  jj/,  2,  .  .  •  t  are  also  subject  to  error. 


§VIIL]      WEIGHTS  OF  THE  UNKNOWN  QUANTITIES.     1 05 
and  if  this  is  substituted  in  the  second,  we  obtain 

y-_324^+     3673 

y  809  *       809  ' 

Finally,  by  the  substitution  of  this  value  of  y  in  the  first  normal 
equation,  we  obtain,  before  any  reduction  is  made, 


19899  x  _  49154 . 
809  '     809  ' 

_  19899    and   „_  49154 

-  ^9899' 


whence 


as  before  found. 

The  Determination  of  the  Measure  of  Precision. 

134.  The  most  probable  value  of  h  in  the  case  of  observations 
of  equal  weight  is  that  which  gives  the  greatest  possible  value 
to  P,  Art,  120,  that  is,  to  the  function 


in  which  the  errors  are  denoted  by  uv  ,  uz  ,  .  .  „  #m,  so  that  we 
may  retain  z>r  ,  z>3  ,  .  .  .  vm  to  denote  the  residuals  which  corres- 
pond to  the  values  of  the  unknown  quantities  derived  from  the 
normal  equations.  By  differentiation  we  derive,  as  in  Art,  69, 
for  the  determination  of  h, 


(i) 


The  value  of  St?  cannot,  of  course,  be  obtained,  but  it  is 
known  to  exceed  2V,  which  is  its  minimum  value,  and  the  best 
value  we  can  adopt  is  found  by  adding  to  2V  the  mean  value 
of  the  excess,  In?  -  Iv\ 

135.  Let  the  true  values  of  the  unknown  quantities  be 
x  +  8xty  +  fy,  .  .  .  /  4-  <5/,  while  x,  y  ,  .  .  .  t  denote  the  values 
derived  from  the  normal  equations.  We  have  then  the  residual 
equations 


1  06  INDIKE  C  T  OBSER  VA  TIONS.  [Art.  i   5 


+  .  .  .  +  lmt—nm=vm 
and,  for  the  true  errors,  the  expressions, 
a£x  +  dx)  +  b^y  +  dy)  +  .  .  .  +  £(/  +  #) 


-    s         a       .(2) 
)  -  nm=  um  J 

Multiplying  equations  (i)  by  ^,  va,  .  .  .  vm  respectively,  and 
adding,  the  coefficient  of  x  in  the  result  is 

«iZ>i  +  azvz  +  .  o  .  +  amvm, 

which  vanishes  by  the  first  normal  equation  (i),  Art.  121.  In 
like  manner,  the  coefficient  of  y  vanishes  by  the  second  normal 
equation,  and  so  on.  Hence 

Ztf=-Zwo  ........  '    (3) 

Treating  equations  (2)  in  the  same  way,  we  have 

2uv  =  —  Znv  ; 
hence 

IV  =  Zwu.       .  .    .         (4) 

Again,  multiplying  equations   (i)   by   ul}  u^,  .  .      um,  and 
adding, 

luv  =  lau  .  x  -f  Ibu  .y  +  .  .  .  +  Zlu  .  i  —  Inu  ; 

and  treating  equations  (2)  in  the  same  way> 

Itf  =  Zau  (x  +  dx)  +  Ibu  (y  +  *y)  +  o  0  .  +  Zlu  (t  +  df}-l'nu 

Subtracting  the  preceding  equation,  we  have,  by  equation  (4), 
IV  —  Zv*  =  lauotx  +  Ibu  .  Sy  +  .  .  .  +  Zlu    8t,    .     (5) 

an  expression  for  the  correction  whose  mean  value  we  are 
seeking. 


§  VIII.]  THE  MEASURE  OF  PRECISION.  IO/ 

136.  Expressions  for  dx,  3y,  .  .  .  St  are  readily  obtained  as 
follows.  Treating  equations  (2)  exactly  as  the  residual  equa- 
tions (i)  are  treated  to  form  the  normal  equations,  we  find 

la?  .  (x  +  dx)  +  lab  .  (y  +  dy)  +  ... 

+  Sal.  (t  +  dt)  =  San  +  Sau 
Sab.(x  +  dx')  +  IP  .  (y  +  dy)  +  ... 

+  Ibl.  (i  +  dt)  =  Ibn  +  Ibu 

Sal.  (x  +  dx)  +  Ibl.  (y  +  dy)  +  .  .  . 

Slu 


Subtraction  of  the  corresponding  normal  equation  from  each 
of  these  gives  the  system, 

Sa*  .  dx  +  lab  .  dy  +  .  .  .  +  Sal.  dt  =  Sau 
Sab  .  dx  +  SP  .  dy  +  .  .  .  -f-  Sbl  .  dt  =  Ibu 

Sal.  dx  +  Sbl  .  dy+  ...  +  SI*  .  dt  =  Slu 

a  comparison  of  which  with  the  normal  equations  shows  that 
8x,  Sy,  .  .  .  dt  are  the  same  functions  of  tfl,  uz,  .  .  .  um  that 
x,  y>  .  .  .  t  are  of  nlt  n2,  .  .  .  n^.  Hence  we  have 


where  04,  a,,  ...  an  have  the  same  meaning  as  in  Art.  130. 

137.  Consider  now  the  first  term,  Sau.dx,  of  the  value  of 
Su*  —  Sv*,  equation  (5),  Art.  135.  Multiplying  the  value  of 
dx  just  found  by 

Sau  =  oiu^  +  a^u*  +  .  .  .  +  flTO#m, 

the  product  consists  of  terms  containing  squares  and  products 
of  the  errors.  We  are  concerned  only  with  the  mean  values  of 
these  terms,  in  accordance  with  the  law  of  facility,  which  is  for 

each  error  -^  e~  2"2.  Since  the  mean  value  of  each  error  is 
zero,  it  is  obvious  that  the  mean  value  of  each  product  vanishes; 


IO8  INDIRECT  OBSERVATIONS.  [Art.  137 

so  that  the  mean  value  of  2  an  .  dx  is  the  mean  value  of 


Now  by  Art.   50  the  mean  value   of  each  of  the  squares 
u\t  u\,  .  .  .  u*m  is  —£  ;  hence  the  mean  value  of  lau  .  8x  is  ^-£  , 

or,  by  equation  (4),  Art.  131,  —r^. 

In  the  same  manner  it  can  be  shown  that  the  mean  value  of 
each  term  in  the  second  member  of  equation  (5),  Art.  135,  is 

-TJ  ;  hence  that  of  Su*  —  2V  is  -^5  ,  and  the  best  value  we  can 
adopt  for  2V  is 


Substituting  this  in  equation  (i),  Art.  134,  we  have 

m  —  /j.  . 

—  whence         /l 


The  Probable  Errors  of  the  Observations  and  Unknown 
Quantities. 

138.  The  resulting  values  of  the  mean  and  probable  error  of 
a  single  observation  are 


=  0.6745      --1          (2) 
•>  —     ' 


m  —  fj.  '  "V  m  — 

and  the  probable  errors  of  the  unknown  quantities  are 
r  r  r 


When  the  observation  equations  have  not  equal  weights  we 


§  VIII.] 


PROBABLE  ERRORS. 


109 


may  replace  2V,  which  represents  the  sum  of  the  squares  ol 
the  residuals  in  the  reduced  equations,  by  Ipv*,  in  which  the 
residuals  are  derived  from  the  original  observation  equations. 
The  formulae  (i)  and  (2)  will  then  give  the  mean  and  probable 
errors  of  an  observation  whose  weight  is  unity. 

It  will  be  noticed  that  when  p.  =  i  the  formulae  reduce  to 
those  given  in  Art.  72  for  the  case  of  one  unknown  quantity. 

139.  Instead  of  calculating  the  values  of  vlt  z>2 , .  .  .  vm  directly 
from  the  residual  equations,  and  squaring  and  adding  the 
results,  we  may  employ  the  formula  for  Iv1  deduced  below. 

By  equation  (3),  Art.  135, 

Ztf  =  -  Znv. 

Now  multiplying  equations  (i)  of  that  article  by  nlt  n^,  .  .  .  Km 
respectively,  and  adding  the  results,  we  have 

Znv=Ian.x  +  Ibn.y  +  .  .  .  +  Iln.t  —  Sn*. 
Therefore 

Zv*  =  In*  -  Zan.x-  Ibn.y-  .  .  .  -  Iln.t.    .     (i) 

The  quantity  In*  which  occurs  in  this  formula  may  be  calcu- 
lated at  the  same  time  with  the  coefficients  in  the  normal  equa- 
tions. It  enters  with  them  into  the  check  equations  of  Art.  127. 

We  may  also  express  2V  exclusively  in  terms  of  these  quan- 
tities, for  if  we  write 

la*      lab  .  .  .  lal    Ian 
lab     IP    .       Ibl    Ibn 


lal     Ibl 

Ian     Ibn 


21*      Iln 

Iln     In* 


and  consider  the  development  of  Dn  in  terms  of  the  elements 
of  its  last  row,  we  see  that 


HO  INDIRECT  OBSERVATIONS.  [Art  139 

where  D,  DXi    .    Dt  have  the  same  meanings  as  in  ArL  128 
hence 


140.  For  example,  in  the  case  of  the  four  observation  equa- 
tions of  Art  115, 

x  —  y  +  22  =    3 
yc  +  2y  —  5*  =    5 

-  x  +  zy  +  3*  =  H 

for  which  the  normal  equations  are  solved  in  Art.  123,  the  value 
of  2ri*  is  671 ;  and  formula  (i)  gives 

>y  =  6?I  _  88  x  49*54  -  70  X  7°6^ 
19899  19899 

1600 


—  107  X 


19899 ' 


in  which  1600  is  the  value  of  Dn.  Substituting  this  value  oi 
JiV  in  the  formulae  of  Art.  138,  we  find 

£  =  0.2836,         r  =  0.1913 

for  the  mean  and  probable  errors  of  an  observation  ;  and  using 
the  weights  found  in  Art.  132,  we  find  for  those  of  the  unknown 
quantities 

ex  =  0.057,         £y  =  °-°77»         e*  =  °-°39> 
rx  =  0.038,        ry  —  0.052,        rz  =  0.026. 

In  this  example  we  have  found  the  exact  value  of  JV;  if 
approximate  computations  are  employed,  the  formula  used  has 
the  disadvantage  that  a  very  small  quantity  is  to  be  found  by 
means  of  large  positive  and  negative  terms,  which  considerably 
increases  the  number  of  significant  figures  to  which  the  work 
must  be  carried.  Thus,  because  2ri*  =  671  in  the  above  exam- 
ple, the  work  would  have  to  be  carried  out  with  seven-place 
logarithms  to  obtain  2V  to  four  decimal  places.  The  direct 


§  VIII. ]  MEASURE  OF  INDEPENDENCE.  1 1 1 

computation  of  the  z/2's  from  the  observation  equations  would 
present  the  same  difficulty  in  a  less  degree* 

141.  Of  course,  no  great  confidence  can  be  placed  in  the 
absolute  values  of  the  probable  errors  obtained  from  so  small  a 
number  of  observation  equations  as  in  the  example  given  above. 
There  being  but  one  more  observation  than  barely  sufficient  to 
determine  values  of  the  unknown  quantities,  the  case  is  com- 
parable to  that  in  which  n  =  2,  when  the  observations  are  direct. 

By  increasing  the  number  of  observations  we  not  only  obtain 
a  more  trustworthy  determination  of  the  probable  error  of  a 
single  observation,  but,  what  is  more  important,  we  increase  the 
weight,  and  hence  the  precision,  of  the  unknown  quantities. 
The  measure  in  which  this  takes  place  depends  greatly  upon 
the  character  of  the  equations  with  respect  to  independence. 
As  already  mentioned  in  Art.  113,  if  there  were  only//,  equa- 
tions it  would  be  necessary  that  they  should  be  independent ; 
in  other  words,  the  determinant  of  their  coefficients  must  not 
vanish,  otherwise  the  values  of  the  unknown  quantities  will  be 
indeterminate.  When  this  state  of  things  is  approached  the 
values  are  ill-determined,  and  this  is  indicated  by  the  small 
value  of  the  determinant  in  question.  The  same  thing  is  true 
of  the  normal  equations.  Aqcordingly,  the  weights  are  small 
when  the  determinant  D  is  small;  thus  the  value  ofD  is  in  a 
general  way  a  measure  of  the  efficiency  of  the  system  of  obser- 
vation equations  in  determining  the  unknown  quantities. 

142.  If  we  write  the  coefficients  in  the  m  observation  equa- 
tions in  a  rectangular  form,  thus, 


J  *>.£-....;  V  .....  4 

the  determinant  D  is,  by  a  theorem  in  determinants,  the  sum  of 
-the  squares  of  all  the  determinants  which  can  be  formed  by 


112  INDIRECT  OBSERVATIONS.  [Art.  142 

selecting  /ji.  columns  of  the  rectangular  array.  The  first  of  these 
determinants  is  that  of  the  coefficients  of  the  first  fj.  equations, 
which,  as  we  have  seen,  vanishes  when  they  are  not  independent, 
and  the  others  are  the  like  determinants  for  all  the  other  com- 
binations of  ^  equations  which  can  be  formed  from  the  m  obser- 
vation equations.  It  follows  that  D  cannot  be  negative,  and 
cannot  vanish  unless  there  is  no  set  of  P.  independent  equations 
among  the  observation  equations. 

143.  By  a  similar  consideration  of  the  values  of  DXt  Dy, .  . . 
Dty  Art.  128,  it  has  been  shown*  that,  for  each  unknown  quan- 
tity, the  value  given  by  the  normal  equations  is  the  weighted 
mean  of  all  the  values  which  could  be  derived  from  p.  selected 
equations,  the  weights  being  the  squares  of  the  corresponding 
determmants.f 

Empirical  or  Interpolation  Formula. 

144.  A  set  of  observation  equations  usually  arises  in  the  fol- 
lowing manner :     One  of  two  varying  quantities  is  a  function  of 
another,  of  known  form,  the  constants  which  occur  having, 
however,  unknown  values.     Simultaneous  values  of  the  varying 
quantities  are  observed.      The  values  of  the  second  quantity 
(the  independent  variable  in  the  functional  expression)   are 
regarded  as  accurate,  and  from  them  are  computed  in  each  case 
the  values  of  the  coefficients  when  the  other  variable  is  treated 
as  a  linear  function  of  the  unknown  quantities.     This  other 
variable  is  then  the  observed  quantity  M  of  our  observation 
equations,  and  the  errors  are  the  differences  between  the  ob- 
served values  and  those  which  accurately  correspond  to  the 
assumed  values  of  the  independent  variable. 

*  J.  W.  L.  Glaisher,  Monthly  Notices  of  the  Royal  Ast.  Soc.,  vol.  xl, 
1880,  p.  607  et  seq. 

!"  When  there  is  but  a  single  unknown  quantity,  say  x,  its  coefficients 
0i,  fla ,  .  .  .  take  the  place  of  these  determinants,  and  the  weight  of  the 
result  is  accordingly  2a2.  Compare  Art.  122.  In  general,  as  between 
two  unknown  quantities,  the  weight  of  that  which  has  the  greater  coeffi- 
cients will  be  the  greater. 


§  V 1 1 1.]     EMPIRICAL  OR  INTERPOLA  TlOtf  FORMULAE.      I  1 3 

Taking  the  two  variable  quantities  as  coordinates,  the  obser- 
vations may  be  represented  by  points,  and  the  problem  before 
us  is  that  of  determining  a  curve  of  known  variety  in  such  a 
manner  as  to  pass  as  nearly  as  possible  through  these  points. 

145.  But  it  may  happen  that,  while  we  know  that  a  functional 
relation  between  the  variable  quantities  exists,  we  have  no  theo- 
retic knowledge  of  the  form  of  the  function.    In  such  cases,  our 
only  resource  is  to  assume  the  form  of  the  function,  being 
guided  therein  by  an  inspection  of  the  points  representing  the 
observations.     An  equation  so  assumed  is  sometimes  called  an 
empirical  formula.     The  constants  involved  in  it  are  deter- 
mined exactly  as  in  the  case  of  formulae  having  a  theoretical 
basis.     The  final  result  can  only  be  judged  of  by  the  residuals. 
If  these  are  numerous  enough,  their  failure  to  follow  the  law  of 
accidental  errors  may  indicate  the  inadequacy  of  the  assumed 
form. 

When  the  formula  as  determined  is  used  to  compute  the 
probable  values  of  the  observed  quantity  corresponding  to  other 
values  of  the  independent  variable,  it  is  called  an  interpolation 
formula.  The  results  can  never  be  satisfactory  except  for 
values  within  the  range  of  the  values  corresponding  to  the 
observations  upon  which  the  formula  is  based. 

Conditioned  Observations. 

146.  We  have  hitherto  supposed  the  unknown  quantities  to 
be  independent  of  one  another,  so  that  any  set  of  simultaneous 
values  is  possible,  and  before  the  observations    all    sets  are 
regarded  as  equally  probable.     It  frequently  happens,  however, 
that  the  unknown  quantities  are  required  to  satisfy  rigorously 
certain  equations  of  condition,  in  addition  to  the  observation 
equations  which  must  be  approximately  satisfied.     The  p  un- 
known quantities  may  thus  be  subject  to  v  equations  of  condition, 
where  v  <  /a,  while  the  whole  number  of  equations  m  +  v  exceeds 
/u.     The  case  may  be  reduced  to  that  already  discussed  by  the 
elimination  of  //  unknown  quantities  from  the  observation  equa- 
tions by  means  of  the  equations  of  condition,  leaving  us  with  m 


1  1  4  INDIRE  C  T  OBSER  VA  TIONS.  [Art  .  1  46 

observation  equations  containing  p  —  v  independent  unknown 
quantities. 

We  shall  consider  only  the  case  (which  is  of  frequent  occur- 
rence) in  which  m  =  /*,  and  the  observation  equations  express 
direct  determinations  of  the  p.  unknown  quantities. 

147.  Let  Ml  ,  M2  ,  .  .  .  M^.  be  the  observed  values  of  X,  Y,  .  .  .  T, 
with  weights  /u/2,  .  .  ,/y,  and  put 


so  that  x,  y,  .  .  .  /  are  the  required  corrections  to  the  observed 
values.  The  equations  of  condition  may  be  reduced  as  in  Art. 
117  to  the  linear  forms 


+  a^     +  ...  +  at  — 

(i) 


The  values  of  x,  y,  .  .  .  t  must  satisfy  these  equations,  which 
are,  however,  insufficient  in  number  to  determine  them,  and,  by 
the  principle  of  Least  Squares,  those  values  are  most  probable 
which,  while  satisfying  equations  (i),  make 


2  +  •  •  •  +/Vf2  =  a  minimum. 
In  other  words,  the  values  must  be  such  that 

p&dx  +p,ydy  +  ..  .  +  pjtdt  =  o,    .    .     .     (2) 

for  all  possible  simultaneous  values  of  dx,  <fy,  .  .  .  dtt  that  is,  for 
all  values  which  satisfy  the  equations, 

a^dx  +  azdy  +  .  .  . 

+  bjy  +  .  .  .  +  b^dt  =  o  |.  j      .     .     .     (3) 


+  fzdy  +  .  .  .  +  f^dt  =  o 
derived  by  differentiating  equations  (i).     Hence,  denoting  the 


§v  in.]  CONDITIONED  OBSERVATIONS.  115 

first  member  of  equation  (2)  by  P  and  those  of  equations  (3) 
by  Si,  Si,  ...  Sr  ,  the  conditions  are  fulfilled  by  values  which 
satisfy  equations  (i)  and  make 

P-hS^-kA-  ...  -kvSv=ot   .    .    .     (4) 

where  &lt  kz,  .  .  .  kv  are  any  constants. 

This  last  equation  will  be  satisfied  if  we  can  equate  to  zero 
the  coefficient  of  each  of  the  differentials,  thus  putting 


,    .    .    .    (5) 


and  this  it  is  possible  to  do  because  we  have  /*  unknown  quan- 
tities and  v  auxiliary  quantities  k^  £2,  .  .  .  £„  which  can  be 
determined  so  as  to  satisfy  the  v  +  /*  equations  comprised  in 
the  groups  (i)  and  (5). 

148.  Substituting  the  values  of  x,y,  .  .  .  /from  equations  (5) 
in  equations  (i),  we  have  a  set  of  linear  equations  to  determine 
the  £'s  which  are  called  the  correlatives  of  the  equations  of  con- 
dition. These  equations  may  be  written  in  the  form 


(6) 


,  ...        v 

in  which  the  summation  refers  to  the  coefficients  of  the  several 
unknown  quantities  ;  thus,  for  example,  2  -r  is  the  sum  of  the 

squares  of  all  the  coefficients  in  the  first  equation  of  condition 
each  divided  by  the  weight  of  the  corresponding  unknown 
quantity.  The  correlatives  being  found  from  these  equations, 


1 1 6  2ND  I  RE  C  T  OBSER  VA  TIONS.  [  A  rt.  1 48 

the  values  of  the  corrections  x,  y,  .  .  .  t  are  given  at  once  by 
equations  (5). 

149.  When  there  is  but  one  equation  of  condition 

a^x  +  a«y  +  .  .  .  +  a^t  =  E, 

the  second  members  of  equations  (5)  reduce  to  their  first  terms, 
and  the  equations  require  that  the  corrections  of  the  several 
unknown  quantities  shall  be  proportional  to  their  coefficients 
in  the  equation  of  condition  divided  by  their  weights.  Equa- 
tions (6)  then  reduce  to  the  single  equation 


and  the  corrections  are 


In  the  very  common  case  in  which  the  numerical  value  of 
each  coefficient  in  the  single  equation  of  condition  is  unity  (for 
example,  when  the  successive  angles  at  a  point,  or  all  the 
angles  of  a  polygon,  are  measured,  or  when  the  sum  of  two 
measured  angles  is  independently  measured),  we  have  the 
simple  rule  that  the  corrections  are  inversely  proportional  to 
the  weights. 

Examples. 

i.  Denoting  the  heights  above  mean  sea  level  of  five  points 
by  X,  Yy  Z,  U,  V,  observations  of  difference  of  level  gave,  in 
feet: 

X  =  573.08         Z  -  Y=  167.33          U-V=  425.00 

Y—X=      2.60         U-Z  —      3.80  V=  319.91 

Y=  575.27         U-  Y=  170.28  y=  319.75 

Putting  X=  573  +  x,  Y=  575  +^,  Z=  742  +  zt  U—  745  +  u. 


§VIIL]  EXAMPLES.  IT? 

V=.  320  +  v,  find  the  values  and  probable  errors  of  the  cor- 
rections x,yy  z,  u,  v,  supposing  the  observations  to  have  equal 
weight. 

x—  —  0.19  ±  0.23,    ^  =  0.14  ±0.21,     z  =  0.05  ±  0.30, 
u  =  0.43  ±  0.25,     v  =  0.03  ±  0.19. 

2.  Given  the  observation  equations : 

•*  =  4-5>  jj/=i.6,  *—.)/  =  2.7, 

with  weights  10,  5  and  3  respectively,  determine  the  values  of 
x  andjy.  x  —  4.468  ±  0.049,  y  =  l-^3  i  0.063. 

3.  Measurements  of  the  ordinates  of  a  straight  line  corres- 
ponding to  the  abscissas  4,  6,  8  and  9,  gave  the  values  5,  8,  10 
and  12.     What  is  the  most  probable  equation  of  the  line  in  the 
formjy  =  mx  +  bl  y  =  1.339.*'  —  0.029. 

4.  Given  the  observation  equations  of  equal  weight : 

x  =  10,  y  —  x  =  7,  y  =  18, 

.r  -  *  =  9.          #  -  *  =  2, 

determine  the  most  probable  values  of  the  unknown  quantities, 
and  the  probable  errors  of  an  observation  and  of  each  unknown 
quantity.  X=IQ§,  ^=17!,  2  =  8%, 

r  =  rz=  0.29,    rx  =  ry  =  0.23. 

5.  In  order  to  determine  the  length  x  at  o°  C.  of  a  meter 
bar,  and  its  expansion  y  for  each  degree  of  temperature,  it  was 
measured  at  temperatures  20°,  40°,  50°,  60°,  the  corresponding 
observed  lengths  being  1000.22,  1000.65,  1000.90  and  1001.05 
mm.  respectively.    Find  the  probable  values  of  x  and  y  with 
their  probable  errors.  x  =  999mm.8o4  ±  0.033, 

y  =  omm.O2i2  ±  0.0007. 

6.  The  length  of  the  pendulum  which  beats  seconds  is  known 
to  vary  with  the  latitude  in  accordance  with  Clairant's  equation, 


where  /'  is  the  length  at  the  equator,  q  the  ratio  ^|^  of  the  cen- 


1  1  8  INDIRECT  OBSER  VA  TIONS.  [Art.  149 

trifugal  force  at  the  equator  to  the  weight,  and  p.  the  compres 
sion  of  the  meridian  regarded  as  unknown.     Putting 


observations  in  different  latitudes  gave  in  millimeters  : 

x  +  0.9697  =  5.13,      x  +  0.0957  =  0.56,      x  +  0.3277  =  1.70, 
x  +  0.7497  =  3.97,      x  —  0.19,      x  +  0.6857  =  3.62, 

x  +  0.4267  =  2.24,      x  +  0.1527  =  0.77,      x  +  0.7937  =  4.23. 

Find  the  length  at  the  equator  with  its  probable  error. 

/'  =  99imm.o69  ±  .026. 

7.  Find  the  value  of  /x  in  the  preceding  example  and  its  prob- 
able error.  /*  =  -^  ±  0.00046. 

8.  The  measured  height  in  feet  of  A  above  O,  B  above  A  and 
B  above  O  are   12.3,  14.1   and  27.0  respectively.      Find  the 
most  probable  value  and  the  probable  error  of  each  of  these 
differences  of  level.      12.5  ±  0.17;  14.3  ±  0.17;  26.8  ±  0.17. 

9.  A  round  of  angles  at  a  station  in  the  U.  S.  Coast  Survey 
was  observed  with  weights  as  follows  : 

65°  n'  52".500  with  weight  3,       87°  2'  24^.703  with  weight  3, 
66    24   15  .553    "          "       3,      141  21    21  .757     "  i; 

find  the  adjusted  values  whose  sum  must  be  360°. 

65°  n'53".4i45>     87°  2'  25".6i75, 
66    24   16  .4675,  141  21    24  .5005. 

10.  Four  observations  on  the  angle  X  of  a  triangle  gave  a 
mean  of  36°  25'  47",  two  observations  on  Y  gave  a  mean  of 
90°  36'  28"  and  three  on  Z  gave  52°  57'  57".    Find  the  adjusted 
values  of  the  angles  and  the  probable  error  of  a  single  obser- 
vation. r  =  7"-7  ;    x  =  36°  25'  44"-23, 

K=  90     36     22   .46, 

^=52    57   53  -31- 

IT.  A  round  of  four  angles  was  observed  as  follows  : 
38°  52'  i4".28  weight  2,      44°  35'  56"«54  weight  3, 
145    23    16  .35      "       4,     131    10   21  .47  3, 

find  the  adjusted  values. 

38°  5i'  35".94»      44°  35'  3o".98, 
145    22  57  .18,     131      9  55  -9i- 


§VIII.]  EXAMPLES. 


12.  Measurements  of  the  angles  between  surrounding  stations 
were  made  with  weights  as  follows  : 

Between  stations  i  and  2,  55°  57'  58".68,  weight  3, 
"  2  "  3,  48  49  13  -64,  "  19, 
"  i  "  3,  104  47  12  .66,  17, 

"       3   "     4,    54    38    15  -53,       "      13, 
2   "     4,  103    27    28  .99,  60 

Pind  the  corrections  of  the  angles  in  the  order  given. 

o".2«5,  o".oo5,  —  o".c»50,  —  o".058,  o".i27, 


IX. 

GAUSS'S  METHOD  OF  SUBSTITUTION. 
The  Reduced  Normal  Equations. 

150.  In  solving  the  normal  equations,  it  becomes  essential, 
except  in  the  simplest  cases,  to  reduce  the  labor  as  much  as 
possible  by  adopting  a  systematic  process  in  the  elimination. 
We  shall  here  give  the  method  of  substitution  as  developed  by 
Gauss,  which  has  the  advantage  of  preserving,  in  each  of  the 
sets  of  simultaneous  equations  which  arise  in  the  elimination, 
the  symmetry  which  exists  in  the  coefficients  of  the  normal 
equations,  thereby  materially  diminishing  the  number  of 
coefficients  to  be  calculated. 

The  m  observation  equations,  involving  the  /*  unknown 
quantities  xt  y,  z,  .  .  .  /,  being,  as  in  Art.  1  24, 

a,x  +  £j>  +...+/,/=  »» 


amx  -\-  bmy  +  .  .  .  -f  lmt  =  nm  , 
let  the  normal  equations  be  written  in  the  form 

\aa~\x  +  [ab]y  +  \ac~\z  +  ...+  \af\t  =  [an] 
\ab\x  +  \bb~\y  +  \bc\z  +  .  .  .  +  \bt\t  =  \bn\ 
\ac\x  +  \bc\y  +  \cc\z  +  .  .  .  +  \ct\t  =  [en] 

[al]X  +  \bl-\y  +  [cl]z  +  ...+  [U]t  =  [In] 

As  mentioned  at  the  end  of  Art.  126,  we  may  suppose  the 
observation  equations  (i)  to  have  been  reduced  to  the  weight 
unity,  so  that  [aa],  \ab\  .  .  .  [In]  stand  for  2a*,  2at>t 


IX.] 


THE  REDUCED  NORMAL   EQUATIONS. 


121 


151.  The  value  of  x  in  terms  of  the  other  unknown  quanti- 
ties derived  from  the  first  of  equations  (2),  or  normal  equation 
for  xy  is 

\ab\          \ac\  .    [an] 

X  ~  ~~  \aa\y  ~~  \aa\Z  ~  '  '  '       Vaa\  ' 

Substituting  this  in  the  /*  —  i  other  equations,  they  become 


-  Mgj)>+  (w  - 


+•  -  -  -  w  -  M 


(M  -  Mg]) 


>+.  . 


-  wgj)  -  w  -  wig) 


in  which  it  will  be  noticed  that  the  coefficients  of  the  unknown 
quantities  have  the  same  symmetry  as  in  the  normal  equa- 
tions (2).  These  equations  for  the  /*  —  i  unknown  quantities 
y,  z,  .  .  .  t  are  called  the  reduced  normal  equations,  and  are 
written  in  the  form 


,  ib  +  !>/, 


[t>I,  i]/  =  [^»,  i 
[^/,  i]/  =  [«i,  i] 

[//,  i]/  =  [/«,  i]  . 


,    -(3) 


in  which 


(4)' 


122  GAUSS'S  METHOD   OF  SUBSTITUTION.  [Art.  151. 

Equations  (4)  show  that  the  rule  for  the  formation  of  the 
coefficients  and  the  second  members  of  the  reduced  normal 
equations  is  the  same  throughout;  namely,  from  the  correspond- 
ing coefficient  in  the  normal  equations  we  are  to  subtract  the 
result  of  multiplying  together  the  two  expressions  in  whose 
symbols  one  of  the  letters  in  the  given  symbol  is  associated 
with  #,  and  dividing  the  product  by  [aa]. 


The   Elimination   Equations. 

152.  Eliminating  y  by  means  of  the  first  of  the  reduced 
normal  equations  (3)  from  each  of  the  others,  just  as  x  was 
eliminated  from  the  normal  equations,  and  employing  a  similar 
notation,  we  have  the  ^  —  2  equations 


[«:,  2>  +  .  .  .  +  |>/,  2]/  =  \cn,  2]  ) 

•   },    •    •     (5) 

|>/,  •]*+...+   [//,  2]/  =  [/«,  2]  ) 

which  may  be  called  the  second  reduced  normal  equations. 
The  coefficients  in  these  equations  are  derived  from  those  in 
equations  (3)  exactly  as  the  latter  were  found  from  those  in 
equations  (2).  Thus 


•     (c 


In  like  manner  the  third  reduced  normal  equations  are 
formed  from  these  last,  the  coefficients  being  distinguished 
by  the  postfixed  numeral  3,  corresponding  to  the  number  of 


§  IX.]  THE   ELIMINATION  EQUATIONS.  123 

variables  which  have   been    eliminated.     We  finally  arrive  at 
the  single  equation 

[//,  /*-!]/  =  [/»,  /I  -    l],       .....   (7) 

which  determines  the  unknown  quantity  standing  last  in  the 
order  of  elimination. 

153.  The  quantity  which  immediately  precedes  /  is  next 
derived  from  the  first  of  the  preceding  set  of  equations  (that 
is,  from  the  equation  by  means  of  which  it  was  eliminated)  by 
the  substitution  of  the  numerical  value  found  for  /  ;  and  so 
on,  until  finally  x  is  found  from  the  first  of  the  original  normal 
equations.  The  equations  from  which  the  unknown  quantities 
are  actually  determined  are  therefore  the  following  : 


\aa~\x  +  \ag\y  +  \ac]z  +  ....+   \_at\t    =  [an] 
\M,  i]y  +  l>>  i>  +  -  •  -  +  W  i]/  =  \bn, 


(8) 


These  are  called  the  finai  or  elimination  equations. 

The   Reduced  Observation   Equations. 

154.  Let  us  suppose  that  there  exists  a  relation  between 
the  variables  which  must  be  exactly  satisfied,  while  the  m 
observation  equations  are  to  be  satisfied  approximately.  Let 
this  relation  be 

Eliminating  x  from  the  observation  equations  (i),  Art.   150 
by  the  substitution  of 

=  -£  -Kg-        --/  +  - 

a         a  a       a* 


124  GAUSS'S  METHOD    OF  SUBSTITUTION.  [Art.  154 

derived  from  this  equation,  we  have 


which   may   be   called   the  reduced  observation   equations,  and 
written  in  the  form 


a  comparison  of  which  with  the  equations  written  above  suffi- 
ciently indicates  the  values  of  £/,  c^'  ,  .  .  .  #w',  .  .  .  nm'. 
The  /^  —  i  normal  equations  derived  from  these  are 


in  which 


?+  \aafj 


-(4) 


§  IX.]     THE   REDUCED    OBSERVATION  EQUATIONS.       12$ 

155.  Let  us  now  suppose  that  the  equation  of  condition 
(i)  which  is  to  be  exactly  satisfied  is  identical  with  the  first 
of  the  normal  equations  (2)  of  Art  150,  so  that 

a  —  [aa],     ft  =  \ab],     ...     v  =  [an] ; 
then  equations  (4)  become 

w 


(s) 


[aa] 

Comparison  of  these  with  equations  (4),  Art.  151,  shows  that 
the  normal  equations  (3)  of  the  preceding  article  now  become 
identical  with  the  first  reduced  normal  equations  of  Art.  151. 
Hence  the  first  reduced  normal  equations  are  the  same  as  the 
normal  equations  corresponding  to  the  reduced  observation  equations 
which  would  result  if  x  were  eliminated  from  the  observation 
equations  by  means  of  the  normal  equation  for  x. 

It  is  evident  that,  in  like  manner,  the  second  reduced 
normal  equations  are  the  same  as  the  yu  —  2  normal  equation 
which  would  result  from  the  reduced  observation  equations,  if 
they  were  further  reduced  by  the  elimination  of  y  by  means 
of  the  reduced  normal  equation  for  y  ;  or,  what  is  the  same 
thing,  the  normal  equations  which  would  result  if  x  and  y  were 
eliminated  from  the  original  observation  equations  by  means 
of  the  normal  equations  for  x  and^.  Similar  remarks  apply  to 
the  other  sets  of  reduced  normal  equations. 

156.  An  important  consequence  of  what  has  just  been 
proved  is  that,  among  the  coefficients  in  the  reduced  normal 
equations,  or  auxiliary  quantities,  those  of  quadratic  form, 

\bb,  ij,     [cc,  i],     ...      \cc>  2],  "...     [//,  »  -i], 


126  GAUSS'S  METHOD   OF  SUBSTITUTION.  [Art.  156. 

being,  like  the  corresponding  quantities  in  the  normal  equa- 
tions, sums  of  squares,  are  all  positive.  It  is  further  to  be 
noticed  that  each  of  these  quantities  decreases  as  its  postfix 
increases,  for  the  subtractive  quantities  in  the  formation  of 
the  successive  values  are  themselves  positive.  For  example. 

\tt  il  -  17/1  -  ^     Ml  2!  -  Ml  il  -  &'  *!' 

L//>I  '   L         L  ' 


Weights  of  the  Two  Quantities  First  Determined. 

157.  The   unknown   quantity   /  has   been   determined   in 
equation  (7),  Art.  152,  after  the  manner  described  in  Art.  133; 
that  is  to  say,  from   its  own  normal  equation  —  no  reduction 
by  multiplication  or  division  having  taken  place  in  the  course 
of  the  elimination.    Hence,  as  proved  in  that  article,  its  weight 
is  the  coefficient  of  the  unknown  quantity;  that  is  to  say,  the 
weight  of  an  observation  being  unity,  that  of  /  is 

/«  =  [//,/<-  i], 

which,  as  shown  in  the  preceding  article,  is  necessarily  a  posi- 
tive quantity.* 

The  weight  of  any  one  of  the  unknown  quantities  might  be 
determined,  in  like  manner,  by  making  it  the  last  in  the  order 
of  elimination. 

158.  Let  s  be  the  unknown  quantity  preceding  /,  so  that 


*  As  shown  in  Art.  156,  the  substitutions  diminish  the  successive  coef- 
ficients of  /.  Compare  the  foot-note  to  Art.  133,  p.  104.  In  fact  [//]  is 
the  weight  that  t  would  have  if  the  true  values  of  all  the  other  quantities 
were  known;  [//,  i]  is  the  weight  which  it  would  have  if  all  the  others 
except  x  were  known—  that  is,  if  x  and  /  were  the  only  quantities  subject 
to  error  ;  and  so  on, 


§  IX.]          THE  REDUCED   EXPRESSION  FOR   iV.  I2/ 


or 


,  p  -  2]  =  [//,  /l  -  2][^,  JJL  -  2]  -  [*/,  yu  -  2]2- 

If  now  the  order  of  s  and  /  be  reversed,  no  other  change  of 
order  being  made,  the  auxiliaries  with  the  postfix  yu  —  2  will 
be  unaltered,  and  we  shall  have 

\kky  V  -   l][//,  /*  -  2]  =    \kk,   »   -    2][//,  }*  ~  2]   -  [*/,  /<  -   2]2, 

hence 

\kk,  /*  —  i][//,  /*  —  2]  =  [//,  M  —  i]|>£,  /^  —  2]. 


But  [/£/£,  /^  —  i]  is  the  weight  of  s,  therefore  we  have 

A  =  [«,  A.  -  1]  =  ffi^r'-,1  [//,  A*  -  -]• 

The  weights  of  the  other  unknown  quantities  cannot  be 
thus  readily  expressed  in  terms  of  the  auxiliaries  occurring 
in  the  calculation  of  /.  A  general  method  of  obtaining  all  the 
weights  will  be  given  in  Arts..  174-176. 

The  Reduced  Expression  for  -2V. 

159.  We  have  found  in  Art.  139  for  2V  or  \vv\  the  expres- 
sion 

[w]  =  -  \an\x  -  \bn~\y  -  ...  -  \ln~\t  +  [«»], 

which  is  similar  in  form  to  the  expressions  equated  to  zero 
in  the  normal  equations.  If  in  this  we  substitute  the  value 
of  #,  as  in  Art.  151,  it  becomes 

[w]  =  —  \bn,  \\y  —  [en,  i]z  —  ...  —  [In,  \]t  +  \nn,  1  1, 


128  GAUSS'S  METHOD   OF  SUBSTITUTION.    [Art.  159 


in  which 


after  the  analogy  of  the  auxiliary  quantities  defined  in  equa- 
tions (4),  Art.  151.  In  like  manner,  by  the  elimination  of 
y,  [mi]  is  reduced  to  the  form 

[w]  =  -  [en,  2>  -  .  .  .  -  [In,  2\t  +  [««,  2], 
and  finally,  by  the  substitution  of  the  value  of  /,  to 
\yv\  =  [nn,  /*], 

the  postfix  n  indicating  that  all  the  unknown  quantities  have 
been  eliminated. 

Substituting  in  the  expressions  for  the  mean  and  probable 
error  of  an  observation,  Art.  138,  we  have 


m—  JA  m  — //• 

The  General  Expression  for  the  Sum  of  the  Squares  of  the 
Errors. 

160.  The  following  articles  contain  an  investigation*  of 
the  sum  of  the  squares  of  the  errors  considered  as  a  function 
of  the  unknown  quantities,  showing  directly  that  the  minimum 

*  Gauss,  "  Theoria   Motus   Corporum   Coelestium,"  Art.  182;    Werke, 
vol.  vii.  p.  238. 


I2Q 

value    of    this    quantity    corresponds   to    the   values    derived 
from  the  normal  equations,  and  is  equal  to  [mi,  //],  and  also 
deriving  from   the   general   expression   the   law   of  facility  of 
error  in  /,  and  thence  its  weight. 
Let 

W=[vv] (i, 

be  the  sum  of  the  squares  of  the  errors  in  the  observation 
equations,  that  is  to  say,  of  the  linear  expressions  of  the  form 
(Art.  119), 

ax-\-  by  -\-  ...  -\-  It  —  n  =  v. 


The  absolute  term  in  W  is  obviously  \nn\.     Put 
idW  idlV 


2  dx  2   dy  2    dt 

Then 


, 

~~ 


X  =  2        =  M  -  \aa\x  +  \_ab\y  +  .  .  .  +  [al]t  -  [an].    (3) 

The    equations   X  =  o,    Y  =  o,  .  .  .  T  =  o   are    the    normal 
equations.     Now,  since 

id(X*)  _       dX  _  i  d   X* 

--  7  —  —  X*~r  —  \aa\X,    or,    —  —  f  —  ^  =  X. 

2     doc  dx  2  dx  [aa] 


1  d 
-7- 

2  dx 


d  I...       X*\ 

-7-   W  —  j  —  ^    =o  ; 

dx\  [aa]J 

hence,  if  we  put 


Wl   is  a  function  independent  of  x.     Now,  in  equation  (4), 
W^  has  for  all  values    of    the  variables  which  make  X  =  o 


130  GAUSS'S  METHOD   OF  SUBSTITUTION.    [Art.  1  60. 

the  same  value  as  W  ';  hence  Wl  is  what  W  becomes  when  x 
is  eliminated  from  it  by  means  of  the  first  normal  equation, 
X=o. 

l6l.  It  follows  from  what  has  just  been  proved,  that 


(5) 


that  is  to  say,  Wl  is  the  sum  of  the  squares  of  expressions  of 
the  form 

*>  +  /*+...  -f  /'/-«'=  z/, 

corresponding  to  the  reduced  observation  equations,  Arts.  154, 
155.     The  absolute  term  in  W^  is  therefore  \n'n']  or  [nn,  i]. 
If,  now,  we  put 


dy  ' 


and  Yl  =  o,  .  .  .  Tl  —  o,  are  the  reduced  normal  equations. 
The    relation    between    the    expressions    Y  ,  ,  ...  T7,    and 
X,  Y,  .  .  .  7"is  derived  from  equation  (4);  thus,  differentiating 
with  respect  to  K, 

.  (8) 


which  gives  another  proof  of  the  identity  of  the  coefficients 
[W],  .  .  .  \b'n'}  with  \bb,  i],  .  .  .  \bn,  i],  established  in  Art.  155. 
We  now  prove,  exactly  as  in  the  preceding  article,  that 


is  a  function  independent  of  y  as  well  as  of  x,  and  is  identical 


§  IX.]   SUM  OF  THE  SQUARES  OF  THE  ERRORS.    \^\ 

with  \v"v"\  the  sum  of  the  squares  of  expressions  of  the 
form 

c"z  +  . . .  +  i"t  _  n»  =  v» 

corresponding  to  the  second  reduced  observation  equations, 
from  which  x  and  y  have  been  eliminated  by  means  of  the 
equations  X  =  o,  Ka=  o.  The  absolute  term  in  W^  is  obviously 
[;;' V']  or  [««,  2]. 

162.  Proceeding  in  this  way,  we  finally  arrive  at  an  expres- 
sion W^  which  is  independent  of  all  the  variables,  and  consists 
simply  of  the  absolute  term  [nn,  /*].  We  have  thus  reduced 
W  to  the  form 


The  denominators  \aa~\,  \bb,  i], .  .  .  [//,  /*  —  i],  being  sums  of 
squares,  are  all  positive;  hence  the  minimum  value  of  W\s  the 
value  [nn,  /*]  corresponding  to  the  values  of  xty,  .  .  .  t  which 
satisfy  the  equations  X  =  o,  Yl  =  o,  .  .  .  T^  _  I  =  o. 

163.  Since  W  is  the  sum  of  the  squares  of  the  errors,  the 
probability  that  the  actual  observations  should  occur  is 
proportional  to  e  ~ h>ilv  as  in  Art.  62.  Therefore,  by  the 
principle  explained  in  Art.  30,  the  observations  having 
been  made,  the  probabilities  of  different  systems  of  values  of 
the  unknown  quantities  are  proportional  to  the  corresponding 
values  of  this  function.  Hence,  C  being  a  constant  to  be 
determined,  the  elementary  probability,  Art.  21,  of  a  given 
system  of  values  of  x,  yt  .  .  .  t  is 

Cc-»wdxdy.  .  .dt,  (n) 


*  This  result  is  also  derived  by  Gauss  in  a  purely  algebraic  manner  in 
the  "  Disquisitio  de  Elementis  Ellipticis  Paladis; "  Werke,  vol.  vi.  p.  22. 
Sej  also  Encke,  Berliner  Astronomisches  Jahrbuch  for  1853,  pp.  273-277. 


132  GAUSS'S  METHOD   OF  SUBSTITUTION.   [Art.  163. 

where  h  is  the  measure  of  precision  of  an  observation,  and  C 
is  such  that  the  integral  of  the  expression  for  all  possible 
values  of  the  variables  is  unity. 

The  probability  of  a  given  system  of  values  of  y,  z,  .  .  .  /, 
while  x  may  have  any  value,  is  found  by  summing  this 
expression  for  all  values  of  x.  It  is  then 


r 

Cdy...dt\_ 


-*•— 


7  "V7" 

since  W^  in  equation  (4)  is  independent  of  x.    Since  —  =  [aa], 

the  value  of  the   definite  integral  in  this  expression  is,  by 
equation  (7),  Art.  39, 


[aa]  J-«  h  \/\aa] 


Thus  the  probability  of   a  given  system   of  values   of  yt 
,  .  .  .  t  is 


(12) 
h 


164.  In  like  manner,  the  probability  of  a  given  system  of 
values  of  z  .  .  .  /,  x  and  y  being  indeterminate,  is 


i*-*C./ 

which,  by  equations  (9)  and  (7),  reduces  to 

-^dz.     .dte-»w*.     .     .     (13) 


£1X.]       PROBABILITY   OF  A    GIVEN    VALUE   OF  t.         133 

Proceeding  in  this  way,  we  have,  finally,  for  the  probability 
of  a  given  value  of  /, 


--.  .    (I4) 
Again,  integrating  this  for  all  values  of  /,  we  have 

ds) 


Substituting  the  value  of  C  thus  determined,  we  obtain  for 
the  probability  of  /, 


I/7T 

But 


and 

TV-  t  =  [//,  /i  -  i]/-  [/«,  ^  -  i]; 

therefore,  putting 

»-i        _          [/«,  /^  -  i] 

"  ' 


and  omitting  dt,  the  expression  (16)  gives  for  the  law  of  facility 
of  error  in  /, 


h          ,       ~  _l]T.  ,     x 


134  (MUSS'S  METHOD    OF   SUBSTITUTION.  [Art.  164. 

This  is  of  the  same  form  as  the  law  of  facility  for  an  ob- 
servation, except  that  the  measure  of  precision  is 

h  |/[//,  /4  -  i]. 

Thus  the  most  probable  value  of  /  is  that*  which  makes 
T  =  o,  namely, 

_  [In,  y  -  i] 

"    [//,/<-!]' 

and  the  weight  of  this  determination,  when  that  of  an  observed 
quantity  is  unity,  is 

Pt  =  [//,  M  ~  i]. 


The  Auxiliaries  Expressed  in  Determinant  Form. 

165.  If,  in  the  determinant  of  the  coefficients  of  the  normal 
equations,  denoted  by  D  in  Art.  128,  we  subtract  from  the 

second  row  the  product  of  the  first  row  multiplied  by  |= — i,  it 

becomes 

o,         \bb,  i],         [fc,  i],     ...     [bl,  i]. 

Treating  the  other  rows  in  like  manner,  the  determinant  D 
is  reduced  to  a  form  in  which  the  first  row  is  unchanged,  and 
the  rest  are  replaced  by  a  column  of  o's  and  the  determinant 
of  the  first  reduced  normal  equations.  Denoting  this  last  de- 
terminant by  D',  we  have  D  =  \ad\D' . 

By  a  similar  reduction  of  D' ,  D  is  further  reduced  to  a  form 
in  which  the  first  two  rows  are  as  in  that  described  above,  and 
the  rest  are  replaced  by  two  columns  of  o's  and  the  determi- 
nant, D",  of  the  second  reduced  normal  equations.  Finally, 
D  is  thus  reduced  to  the  determinant  of  the  elimination  equa- 
tions (8),  Art.  153. 


I X .  ]        A  UXILIA  RIES  IN  DE  TERMIXA  N  T  FORM.  135 


The  successive  forms  of  D  give  the  equations 
D=\aa\D'=.\aa~\\bb,  i]Z?"  =  . .  .  =  \aa\\bb,  i\\cc,  2]  . .  .[//,/<- 1]. 

1 66.  If,  in  the  form  of  D  involving  Z^r),  we  take  the  first  r 
rows,  and  then  any  other  row  (which  will  therefore  be  a  row 
Belonging  to  Z^r)),  the  same  reasoning  shows  that  any  deter- 
minant formed  by  selecting  r  +  i  columns  of  this  rectangular 
block  is  equal  to  the  minor  occupying  the  same  position  in  D. 

We  can  now  express  any  auxiliary,  say  [<*/?,  r\  as  the  quo- 
tient of  two  minors,  of  the  (r  -f-  i)th  and  rth  degree  respec- 
tively, in  D.  This  auxiliary  occurs  in  the  form  of  D  just 
mentioned.  Taking  the  first  r  rows  and  columns  together 
with  the  row  and  column  in  which  the  given  auxiliary  occurs, 
we  have  a  determinant  whose  value  is 

\_aa\bb,  i]  .  .  .  [yy,  r  -  i][«/J,  4 

because  all  the  elements  below  the  principal  diagonal  vanish. 
But  this  determinant  is  equal  to  that  similarly  situated  in  Z>, 
and  the  coefficient  of  [at/3,  r\  is  equal  to  the  determinant 
formed  from  the  first  r  rows  and  columns  of  D.  For  example, 
for  \de,  2]  we  have 


[aa]  [at}  [ae] 
[at]  [M]  [fe] 
[M] 


and 


therefore 


ia\       \ab\          [ae\ 

o        [bS,i]     [be, 

i] 

o             o         \de. 

2] 

1  M       \aS\ 

' 

0           [bb, 

i] 

[aa\     [ab] 

L  e>  2J 

[ab\     [bb} 

[at} 
[»] 


[«] 


[«*] 

[ad}     [bd} 


136  GAUSS'S  METHOD    OF  SUBSTITUTION.   [Art.  167. 

167.  The  same  principle  holds  if  we  include  the  auxiliaries 
involving  the  letter  n,  and  in  particular  the  determinant  Dn  of 
Art.  139  is 

Dn  =  \aa\\bb,  i]  .  .  .  [//,  ;i  -  i][»»,  M]  =  D\nn,  //]; 
therefore 

T  Dn 

[««,//]=—, 

which  is  the  same  value  that  was  found  for  \vv\  on  p.  no. 

Form  of  the  Calculation  of  the  Auxiliaries. 

168.  In  calculating   the   coefficients  which   occur   in   the 
elimination  equations  and  the  value  of  [vv],  it  is  important  to 
arrange  the   work   in    tabular  form,   and   to   apply  frequent 
checks  to  the  computation  to  secure  accuracy.     In  the  an- 
nexed table,*  which  is  constructed  for  four  unknown  quanti- 
ties, the  first  compartment  contains  the  coefficients  and  second 
members  of  the  normal  equations  together  with  the  value  of 
[««],  which  are  derived   from  the  observation   equations,  as 
explained  in  Art.  127.     The  coefficients  are  entered  opposite 
and  below  the  letters  in  their  symbols,  those  below  the  diag- 
onal line,  whose  values  are  the  same  as  those  symmetrically 
situated  above,  being  omitted.     Beneath  those  in  the  first  line 
are  written  their  logarithms,  which  are  used  in  computing  the 
subtractive  quantities  placed  beneath  each  of  the  other  co- 
efficients. 


*  The  tabular  arrangement  is  taken  from  W.  Jordan's  "  Handbuch  der 
Vermessungskunde."  See  also  Oppolzer's  "  Lehrbuch  zur  Bahnbestim- 
mung  der  Kometen  und  Planeten,"  vol.  ii.  p.  340  et  seq.,  where  the  table, 
with  a  somewhat  different  arrangement,  is  given  for  six  unknown  quantities, 
and  an  example  is  fully  worked  out. 


IX.]         CALCULATION   OF    THE  AUXILIARIES. 


M 

log  \aa\ 

[«*] 

log  [a^] 

M 

log  [«<:] 

M 

log  [«</] 

M 

log  [aw] 

M 

log  [«M] 

lo§^6 

M 

^6M 

M 
*JM 

M 
ijW] 

M 

Ab[an] 

M 

4fc\ 

l°g^c 

M 

^CM 

[«/] 

^c[^] 

M 

Ac[_an} 

M 
^J«] 

lQg^d 

Wfl 
^JW] 

\dn\ 
Ad[an] 

M 

^dM 

log^M 

[nn] 
An[an\ 

M 
^.M 

DM,  i] 

log  [66,  i] 

L^.  i] 

log  [6c,  i] 

W  i] 

log  [V,  i] 

[6n,  i] 
log  [6n,  i] 

[^,  i] 

log  [6s,  i] 

log^c 

[«,  i] 
^[^,  i] 

[«/,  I] 

£c[bd,  i] 

[en,  i] 
^c[^,  I] 

j>,  I] 
£o[6s,  i] 

log^d 

[^/,  i] 
B&bd,  i] 

[dn.  i] 
^[to,  I] 

[^,  i] 

jTj[^  i] 

^Bn 

[nn,  i] 
^n[^,  i] 

Jns,  i] 
Jfc'l 

[^   2] 

log  \cc,  2] 

W   2] 

log  [cd,  2] 

M,  2] 

log  [en,  2] 

[«,   2] 

log  [«,  2] 

l°scd 

[^,  2] 
CdW   2] 

[rf»,  a]' 
Cd[cn,  2] 

to.«] 

C  [CS,  2] 

]°g  Cn 

[ww,  2] 

CB[«.,  2] 

[~,  2] 
Cn[cS,   2] 

W  3] 
log  W  3] 

IX«,  3] 
log  [£/»,  3] 

[^,  3] 
log  [ds,  3] 

log  Dn  = 

•  log  / 

[»».  3] 
^B[^.  3] 

[«,  3] 
^.[*,  3] 

/  = 

M  - 

[»«,  4] 

[ns,  4] 

138  GAUSS'S   METHOD   OF  SUBSTITUTION.   [Art.  168. 

In  expressing  the  subtractive  quantities  we  have  adopted 
for  abridgment  the  notation 

Ab=[aa\'      Ac=[w\'      Ad  =  [aa\'      An=\w\' 

The  logarithms  of  these  quantities  are  placed  at  the  side, 
and,  adding  them  successively  to  the  logarithms  above,  the 
antilogarithms  of  the  sums  are  entered  in  their  places.  After 
this  is  done,  the  results  of  subtraction  are  the  auxiliaries  with 
postfix  i,  which  are  to  be  placed  in  corresponding  positions 
in  the  compartment  below. 

In  like  manner  the  third  compartment  is  formed  from  the 
second,  and  in  expressing  the  subtractive  quantities  we  have 
put 


So  also  we  have  put 
and  finally, 


_  [en,  2]  . 

~       • 


-  V"<  3] 


which  is  also  the  value  of  /.  Thus  the  first  four  compartments 
correspond  to  the  several  sets  of  normal  equations,  and  their 
first  lines  to  the  four  elimination  equations.  Finally,  in  the 
fifth  compartment  we  have  computed  \nn,  4],  which  is  the 
value  of  \vv\. 

Check  Equations. 

169.   The  column  headed  s  is  added  for  the  sake  of  the 
check  equations 

[aa]  +  [ab]  +  M  +  M  +  M  + 
[aH]  +  [fit]  +  M  +  W\  +  [M  + 


§  IX.]  CHECK  EQ  UA  TIONS.  1  39 

the  quantities  [as],  .  .  .  [ns]  being  formed  as  in  Art.  127,  ex- 
cept that  we  have  changed  the  sign  of  s,  so  that  for  each 
observation  equation 


The  checks  are  applied  before  the  logarithms  and  sub- 
tractive  quantities  are  entered.  They  require  that  the  algebraic 
sum  of  the  quantities  in  each  line  together  with  those  standing 
above  the  first  term  should  vanish. 

Similar  checks  can  be  applied  in  each  of  the  lower  compart- 
ments. For  example,  if  from  the  second  of  equations  (i)  we 
subtract  the  product  of  the  first  equation  multiplied  by  Ab,  we 
have,  since  Ab[aa\  =  [d$], 

o  +  \bb,  i]  +  \bc,  i]  +  \bd,  i]  +  |>,  i]  +  O,  i]  =  o, 

where  [bs,  i]  has  been  formed  in  precisely  the  same  way  as  the 
other  auxiliaries,  namely, 


In  the  same  manner  we  obtain  the  other  equations  of  the 
group 

\bb,  i]  +  [be,  i]  +  [M  i]  +  \bn,  i]  +  [bs,  i]  =  o  1 

f.    (2) 

\bn<  i]  +  \cn,  i]  +  \dn,  i]  +  \nn,  i]  -f  [«j,  i]  =  o  } 

So  also  we  have  similar  checks  involving  the  auxiliaries 
which  have  the  postfix  2,  and  those  which  have  the  postfix  3, 
and  finally 

[««,  4]  -f  [ns,  4]  =  o. 


140 


GAUSS'S  METHOD   OF  SUBSTITUTION.  [Art.  170. 


a 

b 

> 

c 

d 

i 

n 

t, 

b 
c 
d 

n 
c 
d 
n 

d 

n 
n 

3.1217    .5756 
0.49439  9.76012 

-  .1565 

9ni945i 

—  .0067 
7H826o7 

I.57IO 

o.  19618 

—  5.1050 
on7o8oo 

i 

2-9375 
9.26573   .1061 

.1103 
—  .0289 

—  .0015 

—  .  OO  I  2 

-  .9275 
.2897 

—  2.6943 
-  .9413 

i 

8n70oi2 
7n33i68 
9.70179 

4.1273 
.0078 

.2051 
.0003 

-  .0652 

—  .0788 

—4.2211 
•2559, 

—  i 

4.1328 
.OOOO 

—  .0178 
-  .0034 

—4*3n8 

.0110 

i 

1.3409 
.7906 

—  1.9016 

-2.5692 

—  2 

2.8314 
0.45200 

.1392 
9.14364 

—  .0003 

6n477i2 

—  1.2172 
ono8536 

-1.7530 

oB24378 

I 

8.69164 
6^02512 
9H63336 

4-II95 
.0068 

.2048 
.0000 

.0136 
-  .0598 

—  4-4770 
—  .0862 

I 

4-1328 

.0000 

-  .0144 
.0001 

-4-3228 

.0002 

I 

.5503    .6676 
•5233    .7536 

-I 

8.69720 
8.25157 

4.1127 
0.61413 

.2048 

9.31133 

-0734 

8.86570 

-4.3908 
0,^4254 

I 

4.1328 

.0102 

-  -0145 
.0037 

-4.3230 
—  .2186 

I 

.0270 
.0013 

-  .0860 
-  .0784 

—  I 

7^4490  =  log  / 

4.1226 
0.6I5I7 

—  .0182 
8n26oo7 

—  4.1044 

oB6i325 

O 

.0257 
•oooi 

—  .0076 
.0181 

—  I 

/  =  —  .004415     [vv]  =  .02565 

.0256 

—  -0257 

—  i 

B 


§  IX.J  NUMERICAL   EXAMPLE.  H1 

Numerical  Example. 

170.  As  an  illustration,  let  us  take  the  following  normal 
equations  : 

3.1217.*+    .575^-     -1565*  ~     -oo67/=:     1.5710] 
.5756*  -f-  2.93757  +    .11032-     .0015*  =  —  .9275  I 
-.1565*  +    .1103?  +  4.12730  +    .2051* =  —  .0652  j 
-.0067*—    .00157+    .20512  +  4.1328^  =  —  .0178  J 

together  with 

[mi]  =  1.3409, 

which  were  derived  from  sixteen  observation  equations,  while 
at  the  same  time  the  values  of  [as],  .  .  .  [ns]  were  found  as  in 
the  first  compartment  of  the  table.  The  numbers  in  the  final 
column  are  the  sums  which  should  equal  zero  according  to  the 
check  equations,  the  small  errors  being  due  to  the  rejection  of 
decimals  beyond  the  fourth  place.  The  letters  at  the  side  and 
top  indicate  the  symbol  for  each  auxiliary,  while  the  compart- 
ment gives  the  postfix.  Since  there  are  two  computations  for 
[w],  namely  [«//,  4]  and  —  [ns,  4],  which  agree  within  the 
limits  of  the  uncertainty  of  logarithmic  computation,  we  take 
for  its  value  a  mean  between  them.  Putting  m  =  16  and 
yw  =  4  in  the  formulae  for  e  and  r,  this  value  gives 

e  =  .04623,  r—  .03118, 

for  the  mean  and  probable  error  of  an  observation. 

Values  of  the  Unknown  Quantities  from  the  Elimination 
Equations. 

171.  Dividing  the   elimination   equations,  (8),  Art.  153,  by 
\aa],  \bby  i],  [ct,  2],  \dd,  3],  and  using  the  notation  introduced 
in  Art.  1 68,  they  become 

Acz  +  Adt  =  An  -\ 
BeZ  +  B.it^  Bn  I  , 

*+Cdt=Cn     '       '     '     '     (I' 
t  =  Dj 


142 


GAUSS'S  METHOD    OF  SUBSTITUTION.   [Art.  171. 


The  following  table  gives  the  form  in  which  the  computa- 
tion is  conveniently  arranged,  and  its  application  to  the  ex- 
ample for  which  the  elimination  equations  are  found  in  Art.  170. 


i 

ft 

Q 

^n 

An 

t 
log  t 

z 
log  z 

y 

logy 

X 

—  .004415 

.017847 

.000220 

-.42989 

.00000 
—  .00089 

•50325 
—  .00001 
.00091 

.07943 

-.004415 

.018067 
8.25689 

-.43078 

9^3426 

.58358 

The  weight  of  t  is,  by  Art.  157,  [dd,  3],  and  that  of  z  is,  by 
Art.    158,      JJ  *   [dd,  3]  ;   employing  the  values  computed  in 


Art.  170,  we  have 

log/,  =  0.61517,  log/,  =  0.61305, 

/,  =  4.1226,  pz  —  4.1025  ; 

and  dividing  the  values  of  e  and  /  found  above  by  the  square 
roots  of  the  weights,  we  have  for  /, 

et  =  .02277,  rt=  -01536; 

and  for  z, 

ez  =  .02282,  rz  —  .01539. 

Values  of  the  Unknown  Quantities  Found  Independently. 

172.  In  order  to  obtain  the  general  expressions  for  the 
weights,  it  is  necessary  first  to  express  the  values  of  the  un- 
known quantities  independently  of  each  other.  For  this  pur- 


§  IX.]  COMPUTATION  OF  a,  ,    at,    ETC.  H3 

pose  we  multiply  equations  (i)  of  the  preceding  article  by  i, 
ofj  ,  or,  ,  <^3  ,  respectively,  and  add  the  results,  assuming  the  a  's 
to  be  so  determined  that  the  coefficients  of  y,  z,  and  /  vanish. 
We  shall  thus  have 

x  =  An  +  Bna^  +  Cnat  -f-  Z>ntf,,      .     .     .     ( 
and,  for  *he  determination  of  the  #'s, 


^0  +  ^.«i  -f  «,  =o  !••      ...     (3) 

^d  +  ^"x  +  Qar,  +  «3  =  o  J 

In  like  manner,  to  find  y  we  multiply  the  second,  third  and 
fourth  of  equations  (i)  by  I9fi99  fl99  respectively,  and  add. 
The  result  is 


where  the  ft's  are  determined  by 


Agair,  multiplying  the  last  two  of  equations  (i)  by  i,  ^3,  and 
adding 

z=Cn  +  £>ny3, (6) 

where  y3  is  determined  by 

ct  +  r.  =  o.  •   • (7) 

173'  The  form  for  the  computation  of  alt  <*2,  a^ ,  yS2,  /?s, 
y9,  according  to  equations  (3),  (5),  and  (7),  and  the  numerical 
work  for  the  example  of  Art.  170,  is  as  follows: 


144  GAUSS'S  METHOD   OF  SUBSTITUTION.  [Art.  173 


-<** 

=£, 

:£ 

log  at 

Ioga2 

Ioga3 

.050133 
.009065 

.002146 

—  .000020 
—  .002948 

9n26573 

.059198 
8.77231 

—  .000822 

„ 

"  B 


.000106 
.002448 


.002554 
7.40722 


log  yz  =  8n69720 

The  values  of  ofl ,  y52  and  y3  are  not  found,  as  their  loga- 
rithms only  are  needed. 

We  may  now  recompute  the  values  of  the  unknown  quan- 
tities by  means  of  equations  (2),  (4),  (6)  by  way  of  verifying 
the  values  of  al ,  .  .  .  y3  as  well  as  those  of  #,.../.  The  form 
of  computation  will  be  as  below  : 


An 

k 

&. 

Dn 

X 

y 

* 

t 

.50325 
.07927 
.00106 

-.42989 
—  .00088 

—  .00001 

.017847 

.000220 

-.004415 

.00000 

.58358 

-.43078 

.018067 

-.004415 

The  numerical  values   agree   exactly  with  those  found  in 
Art.  171. 


£  IX.]    WEIGHTS   OF   THE    UNKNOWN  QUANTITIES.    145 


The  Weights  of  the  Unknown  Quantities. 

174.  The  principle  by  which  we  obtain  expressions  for  the 
weights  is  that  proved  in  Art.  132,  namely:  When  the  value  of 
any  one  of  the  unknown  quantities  is  expressed  in  terms  of 
the  second  members  of  the  normal  equations,  its  weight  is  the 
reciprocal  of  the  coefficient  of  the  second  member  of  its  own 
normal  equation;  or  what  is  the  same  thing:  The  reciprocal  of 
the  weight  is  what  the  value  of  the  unknown  quantity  becomes  when 
the  second  member  of  its  own  normal  equation  is  replaced  by  unity 
and  that  of  each  of  the  others  by  zero. 

Restoring  the  values  of  the  quantities  Any  ./?„,  Cn,  Z>n,  the 
values  of  x,  y,  z,  /,  Art.  172,  are 


Equations  (3),  (5),  and  (7),  Art.  172,  show  that  the  values  of 
<*!,...  Xi  are  independent  of  the  values  of  \an\,  \bn\,  \cn\ 
and  \dn\  ;  hence  the  changes  indicated  above,  in  order  to  con- 
vert the  second  members  of  equations  (i)  into  the  expressions 
for  the  reciprocals  of  the  weights,  have  only  to  be  made  in  the 
numerators  [an],  \bn,  i],  [en,  2],  and  \dn,  3],  where,  by  the 
definitions  given  in  Arts.  151  and  152,  we  have,  using  the  no- 
tation of  Art.  1 68, 


146  GAUSS'S  METHOD    OF   SUBSTITUTION.   [Art.  174. 

\bn,  i]  =  \bn\  —  A^an\  ^ 

K  2\  =  \_cn\-Ac\an\-  Be\bn,  i]  L     (2) 

[<///,  3]  =  [</«]  —  Ad[an\  —  B^bn,  i]  —  Cd[<:«,  2]  J 

175.  To  find  the  value  of  —  ,  we  must  now  put  in  the  value 

Px 

of  x 

\an\  =  i,         [£«]  =  o,         [en]  =  o,         \dn\  =  o. 


Making  these  substitutions  and  using  equations  (3),  Art 
172,  the  value  of  \bn,  i]  becomes 

\bn,  i]  =  —  Ab  =  at  ; 
that  of  \cny  2]  then  becomes 

\cn,  2]  =  -  Ac  -  Beal  =  a,  ; 
and  that  of  \dn,  3]  becomes 

\dn,  3]  =  —  ^d  —  ^of,  -  Cd^2  =  a3  . 
Hence  from  the  first  of  equations  (i)  we  infer 


. 

px      \aa\  +  \bb,  i]  ^  [«,  2]  ^  [^,  3]  ' 

176.  Again,  to  obtain  the  weight  of  y,  we  put  in  the  second 
of  equations  (i) 


\an\  =  o,         \bn\  =  i, 


=  o,  </«    =  o. 


These  substitutions  in  equations  (2)  give,  with  the  aid  of 
equations  (5),  Art.  172, 


§  IX  ]    WEIGHTS  OF    THE    UNKNOWN  QUANTITIES.    1 

\bn,  i]  =  i, 

\cn>  2]  =     -JBe  =  ft^ 

\dn,  3]  =  —  Bd 


'rence  we  have 


ft: 


In  like  manner  we  complete  the  system  of  equations 


I           I 

\     a: 

or/             a* 

II 

/*       M 

1  [W,  i] 

i 

i 

/?22              y^32 

A 

0*,  i] 

[^  2]       [^»  3] 

i 

1     [     r3a 

A 

L^J  21     t^  3] 

i 

i 

Pt 

[<#,  3]  J 

(3) 


which  are  readily  extended  to  the  case  of  any  number  of 
unknown  quantities. 

I77»  The  form  of  computation  and  its  application  to  our 
numerical  example  are  given  on  page  148,  the  values  of  the 
logarithms  entered  at  the  top  and  side  being  taken  from  the 
computations  on  pages  140  and  144. 

From  the  logarithms  in  the  last  line  and  log  e2  =  7.32990, 
(e2  =  y1^  \vv\t  p.  140)  we  find  for  the  mean  errors 

ex  =  .02669,     ey  =  .02750,     ez  =  .02283,     et  =  .02277; 
and  hence  for  the  probable  errors 

rr  =  .01800,     ry  =  .01855,     r*  —  -OI539>     rt  =  .01536. 


148 


GAVSS'S  MET/SOD   OF  SUBSTITUTION.   [Art.  177 


log  a* 

log  a: 

log  ft: 

lOg    MS* 

log  ft: 

log  y? 

I 

, 

[aa] 

l 

1'°gTT3  ; 

\bb,  i] 

\bb,  i] 

°g  [M,  i] 

&: 

ft: 

i 

i 

\CCy       2] 

\P,  »i 

[«,  2] 

°g  \cc,  2] 

a* 

ft: 

i/ 

I 

W  3] 

IM,  3] 

I  </</.  ^  i 

g  [^,  3] 

I 

i 

i 

I 

A 

A 

A 

pt 

i 

Io£  — 
Px 

logi 

logA 

^J, 

8.53146 

7.54462 

7.38328 

3.82974 

4.81444 

7.39440 

•32034 

9.50561 

.01201 

•35318 

9.54800 

.00085 

.00059 

•24315 

9.38587 

.OOOOO 

.OOOOO 

.00060 

9-38483 

.33320 

•35377 

.24375 

9.52270 

0.54872 

9.38694 

9.38483 

§  IX. J  EXAMPLES.  149 

Examples. 

1.  Show  that  the  values  of  pz  when  there  are  four  unknown 
quantities  given  in  Arts.  158  and  176  are  identical. 

2.  Show  that  the  weight  of  the  determination  of  \bn\  is  \bb\, 
that  of  \bn,  i]  is  \bb,  i],  and  so  on. 

3.  Show  that,  if  the  normal  equation  for  x  were  known  to 
be  exactly  true,  the  values  of  the  unknown  quantities  and  the 
weights  relatively  to  that  of  an  observation  of  all  except  x 
would  be  unchanged,  and  that  the  weight  of  an  observation 
would  be  increased  in  the  ratio  m  —  ft  -f-  i  :  m  —  ju. 

4.  Solve  the  following  normal  equations  which  resulted  from 
twelve  observation  equations  : 

5.1143*  -     0.2792);  -f  3.34602  =  —  0.7365, 

—  0.2792*  -f-  14.6142^  -f-  0.19582  =       2.1609, 

3.3460*  +    0.19587  +  7-6754-s  =  —  0.8927, 

\nn\  =       0.5379, 

and  find  the  probable  errors  of  the  unknown  quantities. 

x   =—.0803,      y   =.i475»      z   =.0851; 
rx  =       .034,        ry  =  .017,        rz  =  .028. 

5.  Solve  the  normal  equations 

5.2485*  -  1.7472?  -  2.1954^  =  -  0.5399, 

—  1.74727  -f-  1.88597  +  0.80412  =   i.4493> 

—  2.19547  -f-  0.80417 -{-  4.04402  =   i. 8681, 

\nn\  =   2.6322; 

and  given  m  =  10,  find  the  probable  errors. 

\vv\  —  0.5504,       *   =  0.422,    y  —  0.945,     2  =  0.503; 
r  =  0.189,          7-3,  =  0.108,     rv  =  0.166,     rz=  0.107. 


ISO  GAUSS'S  METHOD   OF  SUBSTITUTION.  [Art.  177. 

6.  Show  that  the  observation  equations 

0.707.* -|-  2.0527  —  2.3722  —  o.22i/  =  —  6.58, 
0.471*  +  1.3477  —  1-715*  -  o.o85/  =  -  1.63, 
0.260*  +  0.770);  —  0.3562  +  0.483^  =  4.40, 

0.092*  +  0.3437  +  0.2352  +  0.469^  =  10.21, 

0.414*  -f-  1.2047  —  1.5062  —  0.2O5/  =  —  3.99, 
0.040*  +  0.1507  +  0.1042  -+•  o.2o6/  =  4.34, 

give  rise  to  the  normal  equations 

0.971*4-2.8217--    3.1753  —  o.i04/=  —    4.815, 

2.821*  -f-  8.208^  —    9.1682:  —  o.25i/  =  —  12.961, 

"~"  3^7  5X  —  9.1687  +  11.0282  +  O-938/  =       25.697, 

—  0.104*  —  0.2517+   0.9380  +  o-594/  =       10.218, 


and  to  \nn\  =  204.313.     Determine  the  unknown  quantities 
and  the  probable  errors  of  an  observation. 

x  =  —  86.41,  7  =  25.18,  2  =  —  3.12,  /  =  17.66,  r  =  i. 80. 

7.  Account  for  the  small  values  of  the  weights,  especially  of 
*  and  7,  in  Ex.  6.     Show  directly  from  the  value  of  \bb,  i]  that 
/„  <  .012  and/j  <  .0015. 

8.  Ten  observation  equations  gave  the  normal  equations 

2.02530*  -f-  0.638097  —    3.992852  =  —  30.466, 

0.63809*4-0.216497 —    1.120892=  —  11.959, 

—  3.99285*  —  1.12089;'  "i"  10.000002  =  —    6.000, 

together  with  \nn\  =  24928.;  find  the  values  and  weights  of  the 
unknown  quantities  and  the  probable  errors. 


§  IX.]  EXAMPLES. 


x  =  -  202.8,     y   =    286.3,    z  =  —  49.5; 
/*=      .0314,    /W  =  .oo66,    /*  =      .9119; 
r  =  37.702,     *•*  =        213,       rv=  463,       r2  =       39. 

9.  Given  the  following  observation  equations  of  equal  weight: 

.986^  +  .056)'  =  .000,  -953^  +  .1827  =       i.  060, 

•973*  +  .1037  =  -53°:  .943*  +  .  2197=  —    .380. 

.968^  +  .I2$y  =  .680,  .919.*;  +.3077—  .200, 

•959-*  +  •I577  =  -200,  .916*  +  .3177  =  —    .530, 

.912.*  +  3317  =  .000, 

find  the  normal  equations  and  the  value  of  \nn\  by  the  method 
of  Art.  127.  (Notice  that  when  we  put  ^H-^  +  ^  +  'f^oas 
in  Art.  169  a  considerable  saving  of  labor  results  from  the 
fact  that  2(a  +  6)*  =  2(n  +  *)3,  etc.) 

8.08843:  -f  1.67987  =  1.7160, 

1.6798^+  0.43837  =  0.1725, 

\nn\  =  2.3722. 

10.  Solve  the  normal  equations  found  in  Ex.  9. 

x  —  0.642,    y  =  —  2.07,     rx  =  0.25,     rv  =  1.09. 

11.  Thirteen  observation  equations  give  the  normal  equa« 

tions 

17.50*  —    6.507  —    6.502  =         2.14, 

—  6.50^+17.507—    6.502=        13-96, 

—  6.503:  —    6.507  -j-  20.502  =  —    5.40, 

[nn\  —      100.34; 

find  the  values  and  probable  errors  of  the  unknown  quantities, 
x  =  0.67  ±  0.60,    y  =1.17  ±0.60,     z  =0.32  ±0.55- 


2  GAUSS'S  METHOD   OF  SUBSTITUTION.     [Art.  177 

12.  Solve  the  normal  equations 


459*- 

—  3°8*  4~  464?  4~4o&s  —  269^  =  —    695, 

—  389*4-408^4-676*  —  33 1/  =-•    653, 
.244*  —  2697  —3312  4-  469*  =         283> 

\nn\  =       1129. 

x  =  — 0.212, y  =  — i«47i>  2  =  — 0.195,  '  ==  — 0.488; 

C^«j/««/J. 

p  =  0.4769352,  log  p  =  9.6784603  ; 

Pi/2  =  0.6744897,      log  Pi/2  =  9.8289753  ; 
PI/7T  =  0.8453475,     log  Pi/?z^=  8.9270353  ; 

r  =  pVT .  e  =  pV^  .  77. 
Note   that  Pi/2  =  «4-^4-r  +  ^  +  --->  where  a  =  | 


VALUES  OF  THE  PROBABILITY  INTEGRAL, 

OR  PROBABILITY  OF  AN  ERROR  NUMERICALLY  LESS  THAN  X. 

TABLE  I.— VALUES  OF  Pt . 


--—,\  r-*=-^Erf<. 


<!  ° 

i 

2 

3 

4 

5 

6 

7 

8 

9 

0.0 
0.1 
0.2 

o.oooo 
1125 

2227 

0.0113 
1236 
2335 

O.O226 
1348 
2443 

0.0338 
H59 
2550 

0.0451 
1569 
2657 

0.0564 
1680 
2763 

0.0676 
1790 
2869 

0.0789 
1900 
2974 

0.0901 
2009 
3°79 

0.1013 
2118 
3183 

o-3 

0.4 

o-5 

0.3286 
4284 

52°5 

0.3389 
4380 
5292 

0.3491 

4475 
5379 

0-3593 
4569 

5465 

0.3694 
4662 
5549 

0-3794 
4755 
5633 

0-3893 
4847 
57i6 

0.3992 
4937 
5798 

0.4090 
5027 
5879 

0.4187 
5^7 
5959 

0.6 
0.7 

0.8 

0.6039 
6778 
7421 

0.6117 
6847 
7480 

0.6194 
6914 
7538 

0.6270 
6981 
7595 

0.6346 
7047 
7651 

0.6420 
7112 
7707 

0.6494 

7i75 
7761 

0.6566 
7238 
7814 

0.6638 
7300 
7867 

0.6708 

7361 
7918 

0.9 
.0 
.1 

0.7969 
8427 
8802 

0.8019 
8468 
8835 

0.8068 
8508 
8868 

0.8116 

8548 
8900 

0.8163 
8586 
8931 

0.8209 
8624 
8961 

0.8254 
8661 
8991 

0.8299 
8698 
9020 

0.8342 

8733 
9048 

0.8385 
8768 
9076 

.2 

•3 
«4 

0.9103 
9340 
9523 

0.9130 
9361 
9539 

o.9'55 
938i 
9554 

0.9181 
9400 
9569 

0.9205 
9419 
9583 

0.9229 
9438 
9597 

0.9252 
9456 
9611 

0.9275 

9473 
9624 

0.9297 
9490 
9637 

0.9319 

95°7 
9649 

".6 

•  7 

0.9661 

9763 
9838 

0.9673 
9772 
9844 

0.9684 
9780 
9850 

0.9695 
9788 
9856 

0.9706 
9796 
9861 

0.9716 
9804 
9867 

0.9726 
9811 
9872 

0.9736 
9818 
9877 

0.9745 
9825 
9882 

0-9755 
9832 
9886 

.8 
•9 

2.0 

0.9891 
9928 
9953 

0.9895 
993  * 
9955 

0.9899 
9934 
9957 

0.9903 
9937 
9959 

0.9907 

9939 
9961 

0.9911 
9942 
9963 

0.9915 

9944 
9964 

0.9918 

9947 
9966 

0.9922 

9949 
9967 

0.9925 

995i 
9969 

2.1 

2.2 
2-3 

0.9970 
9981 
9989 

0.9972 
9982 
9989 

0.9973 
9983 
999° 

0.9974 
9984 
999° 

0.9975 

9985 
9991 

0.9976 
9985 
999  l 

0.9977 
9986 
9992 

0.9979 

9987 
9992 

0.9980 
9987 
9992 

0.9980 
9988 
9993 

2.4 
2.5 
2.6 

2-7 
2.8 

0.9993 
9996 
9998 

0.9993 
9096 
9998 

0-9994 
9996 
9998 

0.9994 
9997 
9998 

0.9994 
9997 
9998 

0-9995 
9997 
9998 

0.9995 
9997 
9998 

0-9995 
9997 
9998 

0.9995 

9997 
9998 

0.9996 
999s 
9999 

0.9999 

i  .0000 

... 

... 

TABLE  II. — VALUES  OF  Pz. 


x    t   r>    2  17        £     —  H 

/•    />  '   z    V*       ~~  *J  -  J0 

z 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

0.0 
O.I 
0.2 

0.0000 

0538 
1073 

0.005^ 

°59T 
1126 

0.0108 
0645 

1180 

0.0161 
0699 
1233 

0.0215 
0752 
1286 

0.0269 
0806 
1339 

0.0323 
0859 
1392 

0.0377 
0913 
1445 

0.0430 
0966 
1498 

0.0484 

IO2O 
1551 

0.4 
0.5 

0.1604 
2127 
2641 

0.1656 
2179 
2691 

0.1709 
2230 
2742 

0.1761 
2282 
2793 

0.181^1 

0.1866 
2385 
289: 

0.1919 
2436 
2944 

0.1971 
2488 
2994 

0.2023 
2539 
3°44 

0.2075 
2590 
3093 

0.6 
0.7 
0.8 

3632 

4105 

0.3192 
3680 

0.3242 
3728 
4198 

0.3291 

3775 
4244 

4290 

0.3389 
3871 
4336 

0.3438 
39l8 
4381 

0.3487 

3965 
4427 

0-3535 
4012 
4472 

0.3584 
4059 

4517 

0.9 
.0 
.1 

0.4562 
5000 
5419 

0.4606 

5043 
5460 

0.4651 
5085 

0.4695 
5128 
5540 

0-4739 
5170 
5581 

0.4783 
5212 
5621 

0.4827 
5660 

0.4871 

5295 
5700 

0.4914 
5337 
5739 

0.4957 
5378 

5778 

.2 
•3 

-4 

0.5817 
6194 
6550 

0.5856 
6231 
6584 

0.5894 
6267 
6618 

0.5932 
6303 
6652 

o.597i 
6339 
6686 

0.6008 

6375 
6719 

0.6046 
6410 
6753 

0.6083 
6445 
6786 

0.6121 
6480 
6818 

0.6157 

6515 
6851 

.*6 

-7 

0.6883 
7485 

0.6915 

7225 
7512 

0.6947 

7255 
7540 

0.6979 
7284 
7567 

0.7011 
7313 
7594 

0.7042 

7343 
7621 

0.7073 

73/1 
7648 

0.7104 
7400 
7675 

0.7134 
7428 
7701 

0.7165 

7457 
7727 

.8 
•9 

2.0 

0-7753 
8000 
8227 

0.7778 
8023 
8248 

0.7804 
8047 
8270 

0.7829 
8070 
8291 

0.7854 
8093 
8312 

0.7879 
8116 
8332 

0.7904 
8138 
8353 

0.7928 
8161 
8373 

0.7952 
8183 

8394 

0.7976 
8205 
8414 

2.1 
2.2 
2-3 

0.8433 
8622 
8792 

0.8453 
8639 
8808 

0.8473 
8657 
8824 

0.8492 
8674 
8839 

0.8511 
8692 
8855 

0.8530 

8709 
8870 

0.8549 
8726 
8886 

0.8567 

8743 
8901 

0.8585 
8759 

8916 

0.8604 
8776 
8930 

2.4 
2-5 
2.6 

0.8945 
9082 
9205 

0.8959 

9095 
9217 

0.8974 
9108 
9228 

0.8988 
9121 
9239 

0.9002 

9r33 
9250 

0.9016 
9146 
9261 

0.9029 

9158 
9272 

0.9043 
9170 
9283 

0.9056 
9182 
9293 

0.9069 
9194 
93°4 

2.7 

2.8 

2.9 

0.9314 
94" 
9495 

0.9324 
9419 
9503 

0-9334 
9428 

95" 

0.9344 
9437 
9S19 

0-9354 
9446 
9526 

0.9364 
9454 
9534 

0.9373 
9463 
954i 

0-9383 
9548 

0.9392 

9479 
9556 

0.9401 
9487 
9563 

3-2 

o.957o 

9635 
9691 

0.9577 
9641 
9696 

0-9583 
9647 
9701 

o.959o 
9652 
9706 

0-9597 
9658 

97" 

0.9603 
9664 
9716 

0.9610 
9669 
9721 

0.9616 

9675 
9726 

0.9622 
9680 
9731 

0.9629 
9686 
9735 

3-3 
3-4 
3- 

0.9740 
9782 
9570 

3.9744 
9786 

9635 

0.9749 
9789 
9691 

0-9753 
9793 
9740 

0-9757 
9797 
9782 

0.9762 
9800 
9818 

0.9766 
9804 
9848 

0.9770 
9807 
9874 

0.9774 
9811 
9896 

0.9778 
9814 
99'5 

4- 

1: 

0.9930 
9993 
9999 

3.9943 
9994 
.0000 

0.9954 
9995 

0.9963 
9996 

0.9970 
9997 

0.9976 
9998 

0.9981 
9998 

0.9985 
9999 

0.9988 
9999 

0.9991 
9999 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

I 

2 

3 
4 
5 

I 
4 

.1 

25 

I 

8 

% 

125 

1.0000 
I.4I42 
I.732I 
2.0000 
2.2361 

I.OOOO 
1.2599 
1.4422 

1.5874 
I.7IOO 

6 

I 

9 

10 

36 
49 
64 
81 

I  00 

216 

343 
512 

729 

I  OOO 

2.4495 
2.6458 
2.8284 
3-0000 
3.1623 

1.8171 
1.9129 
2.OOOO 
2.O8OI 
2.1544 

ii 

12 
13 
H 
15 

I  21 

i  44 
169 
i  96 

2  25 

i  33J 
i  728 

2  197 

2  744 
3  375 

3.3166 
3.4641 
3.6056 
3-74I7 
3.8730 

2.224O 
2.2894 

2.3513 
2.4IOI 
2.4662 

16 

17 

18 

19 
20 

2  56 

2  89 

3  J4 
3  61 
4  oo 

4  096 

4  913 
5832 
6  859 

8  ooo 

4.OOOO 
4.I23I 
4.2426 

4.3589 
4.4721 

2.5198 

2.5713 
2.6207 
2.6684 
2.7144 

21 
22 

23 
24 

25 

4  4i 
4  84 
5  29 
576 
6  25 

9  261 
10  648 

12  167 

13  824 
15  625 

4.5826 
4.6904 

4.7958 
4.8990 
5.OOOO 

2.7589 
2.802O 
2.8439 
2.8845 
2.9240 

26 

27 
28 
29 
30 

6  76 
7  29 
7  84 
841 

9  oo 

17  576 
19  683 

21  952 
24  389 

27  ooo 

5.0990 
5.1962 

5.2915 

5.3852 
5.4772 

2.9625 
3.0000 
3.0366 
3.0723 
3.1072 

31 
32 

33 
34 
35 

9  61 
10  24 
10  89 
ii  56 

12  25 

29  791 
32  768 

35  937 
39  304 
42  875 

5.5678 
5.6569 
5.7446 
5.8310 
5.9161 

3-MI4 
3«I748 
3.2075 
3.2396 
3.2711 

36 

% 

39 

40 

12  96 
13  69 

14  44 

I|  21 

16  oo 

46  656 
50  653 
54  872 

59  3*9 
64  ooo 

6.0000 
6.0828 
6.1644 
6.2450 
6.3246 

3-30I9 
3.3322 
3.3620 
3.3912 
3.4200 

4i 
42 

43 
44 
45 

16  81 
17  64 
18  49 
19  36 

20  25 

68  921 
74  088 
79  507 
85  184 
91  125 

6.4031 
6.4807 

6-5574 
6.6332 

6.7082 

3.4482 
3.4760 
3-5034 
3-5303 
3.5569 

46 

47 
48 

49 

5° 

21  l6 
22  09 

23  °4 
24  oi 
25  oo 

97  336 
103  823 
no  592 
117  649 
125  ooo 

6.7823 

6.9282 
7.0000 
7.0711 

3-5830 
3.6088 
3-6342 

3-6593 
3.6840 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

51 
52 

53 
54 
55 

26  01 

27  04 
28  09 

29  1  6 
30  25 

I32  65I 
140  608 
I48  877 
157  464 

166  375 

7.1414 
7.2III 
7.2801 

7.3485 
7.4162 

3.7084 
3-73^5 

3-7563 
37798 

3.8030 

56 

H 
S 

3i  36 

32  49 
33  64 
34  81 
36  oo 

175  616 

185  193 
195  112 

205  379 
216  ooo 

7.4833 
7.5498 
7.6158 
7.68II 
7.7460 

3-8259 
3-8485 
3.8709 

3-8930 

3-9  i  49 

61 
62 

P 
64 

65 

37  21 
38  44 
39  69 
40  96 
42  25 

226  981 
238  328 
250  047 
262  144 
274  625 

7.8102 
7.8740 
7-9373 

8.0000 
8.0623 

3-9365 
3-9579 
3-9791 
4.0000 
4.0207 

66 

67 
68 
69 
70 

43  56 
44  89 
46  24 
47  61 

49  oo 

287  496 
300  763 

314  432 
328  509 

343  ooo 

8.1240 
8.1854 
8.2462 
8.3066 
8.3666 

4.0412 
4.0615 
4.0817 
4.1016 
4.1213 

7i 

72 

73 
74 
75 

5°  4i 
5i  84 
53  29 
54  76 
56  25 

357  9" 
373  248 
389  017 
405  224 
421  875 

8.4261 

8.4853 
8.5440 
8.6023 
8.6603 

4.1408 
4.1602 
4.1793 
4.1983 
4.2172 

76 

77 
78 

79 

80 

57  76 
59  29 
60  84 
62  41 
64  oo 

438  976 
456  533 
474  552 
493  °39 
512  ooo 

8.7178 

8-775° 
8.8318 
8.8882 
8-9443 

4-2358 
4.2543 
4.2727 
4.2908 
4.3089 

Si 
82 
83 
84 
85 

65  61 
67  24 

68  89 
70  56 
72  25 

S31  44i 
55i  368 
57i  787 
592  704 
614  125 

9.0000 

9-0554 
9.1104 
9.1652 
9.2195 

4.3267 

4-3445 
4.3621 

4.3968 

86 

87 
88 

89 
90 

73  96 
75  69 
77  44 
79  21 
81  oo 

636  056 

658  5°3 
681  472 
704  969 

729  ooo 

9.2736 

9-3274 
9.3808 
9.4340 
9.4868 

4.4140 
4.4310 
4.4480 
4.4647 
4.4814 

9i 
92 
93 
94 

95 

82  81 
84  64 
86  49 
88  36 
90  25 

753  57i 
778  688 

804  357 
830  584 

857  375 

9-5394 
9-59*7 
9-6437 
9.6954 

9.7468 

4.4979 
4-5J44 

4.5307 
4.5468 

4.5629 

96 

97 
98 

99 

100 

92  16 
94  09 

96  04 
98  01 

I  OO  OO 

884  736 
912  673 
941  192 
970  299 

I  COO  OOO 

9.7980 
9.8489 
9.8995 
9.9499 

I  O.OOOO 

4.5789 
4-5947 
4.6104 
4.6261 
4.64  16 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

IOI 
102 
I03 
104 
105 

O2  OI 
04  04 
06  09 

08  16 
10  25 

I  030  301 
06  I  208 
092  727 
124  864 
157  625 

10.0499 
10.0995 
10.1489 
10.1980 
10.2470 

4.6570 
4.6723 

4-6875 
4.7027 

4.7177 

1  06 
107 
1  08 
109 

no 

12  36 

14  49 
16  64 
18  Si 

21  OO 

191  016 

225  043 
259  712 
295  029 
331  ooo 

10.2956 
10.3441 
10.3923 
10.4403 
10.4881 

4.7326 

4-7475 
4.7622 
4.7769 
4.7914 

III 
112 

i*3 

114 

"5 

23  21 

25  44 
27  69 
29  96 

32  25 

367  631 
404  928 
442  897 
481  544 
520  875 

10-5357 
IQ-5830 
10.6301 
10.6771 

10.7238 

4.8059 
4.8203 
4.8346 
4.8488 
4.8629 

116 
117 
118 
119 

120 

34  56 
36  89 

39  24 
41  61 

44  oo 

560  896 
601  613 
643  032 
685  159 
728  ooo 

10.7703 
10.8167 
10.8628 
10.9087 
10.9545 

4.8770 
4.8910 
4.9049 
4.9187 
4.9324 

121 

122 

123 

124 

125 

46  41 
4884 
51  29 
53  76 
56  25 

771  561 
815  848 
860  867 

906  624 
953  J25 

II.OOOO 

11.0454 
11.0905 
".1355 

11.1803 

4.9461 

4-9597 
4-9732 
4.9866 

5.0000 

126 
127 
128 
I29 
I30 

58  76 
61  29 
6384 
66  41 
69  oo 

2  000  376 
2  048  383 
2  097  152 
2  146  689 
2  197  000 

11.2250 

11.2694 
11.3137 
11.3578 

11.4018 

5-oi33 
5.0265 

5-0397 
5.0528 
5.0658 

*3l 
132 

133 

134 
*35 

71  61 
74  24 
7689 

79  56 
82  25 

2  248  091 
2  299  968 
2  352  637 
2  406  104 

2  460  375 

11-4455 
11.4891 
11.5326 

U.5758 
11.6190 

5.0788 
5.0916 

5-I045 
5.1172 
5.1299 

136 

137 
138 

139 
140 

8496 
87  69 
90  44 
93  21 
96  oo 

2  515  456 

2  57i  353 

2  028  O72 
2  685  619 

2  744  ooo 

11.6619 
11.7047 

n.7473 
11.7898 
11.8322 

5.1426 

5-'55* 

5.1676 
5.1801 
5-1925 

141 

142 

M3 

144 

us 

i  98  81 

2  01  64 

2  04  49 

2  07  36 

2  IO  25 

2  803  221 
2  863  288 
2  924  207 
2  985  984 

3  048  625 

11.8743 
11.9164 
".9583 

I2.OOOO 
I2.O4I6 

5.2048 
5.2171 
5-2293 
5.2415 
5-2536 

146 

J47 
148 
149 

!5° 

2  13  16 
2  16  09 
2  19  04 

2  22  01 

2  25  OO 

3  112  136 
3  176  523 
3  241  792 
3  3°7  949 
3  375  ooo 

12.0830 
12.1244 

I2-l655 
12.2066 
12.2474 

5.2656 
5.2776 
5.2896 

5.3015 
5-3I33 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

IS' 

152 
153 
154 
155 

2  28  01 
2  31  04 

2  34  09 
2  37  16 

2  40  25 

3  442  951 
3  511  808 
3  58i  577 
3  652  264 
3  723  875 

12.2882 
12.3288 
12.3693 
12.4097 
12.4499 

5.325I 
5.3368 

5-3485 
5-360I 
5-37I7 

IS6 

*57 
158 

2  43  36 
2  46  49 
2  49  64 
2  52  81 

2  56  00 

3  796  416 
3  869  893 
3  944  312 
4  019  679 
4  096  ooo 

12.4900 

12.5300 
12.5698 
12.6095 
12.6491 

5.3832 

5-3947 
5.4061 

5-4175 
5.4288 

161 
162 

'63 
164 
165 

2  59  21 
2  62  44 

2  65  69 

2  68  96 

2  72  25 

4  173  281 
4  251  528 

4  33°  747 
4  410  944 
4  492  125 

12.6886 
12.7279 
12.7671 
12.8062 
12.8452 

5.4401 

5-4514 
5.4626 

5-4737 
5.4848 

1  66 
167 
1  68 
169 

170 

2  75  56 

2  78  89 
2  82  24 

2  85  61 

2  89  OO 

4  574  296 
4  657  463 
4  74i  632 
4  826  809 
4  913  ooo 

12.8841 
12.9228 
12.9615 
13.0000 
13.0384 

5-4959 
5.5069 
5.5178 
5.5288 
5-5397 

171 
172 
173 
174 
175 

2  92  41 

2  95  84 
2  99  29 
3  02  76 
3  06  25 

5  ooo  211 
5  088  448 

5  177  717 
5  268  024 

5  359  375 

13.0767 
13.1149 

13-1529 
13.1909 
13.2288 

5-5505 
5-5613 
5-5721 
5.5828 
5-5934 

176 
177 
178 

179 
180 

3  °9  76 
3  13  29 
3  16  84 
3  20  41 

3  24  oo 

5  451  776 
5  545  233 
5  639  752 
5  735  339 
5  832  ooo 

13-2665 
1  3-  3°4i 
I3-34I7 
I3.379I 
13.4164 

5.6041 

5.6i47 
5.6252 

5.6357 
5.6462 

181 

182 

183 
184 
185 

3  27  61 
3  3i  24 
3  34  89 
3  38  56 
3  42  25 

5  929  74i 
6  028  568 
6  128  487 
6  229  504 
6  331  625 

I3-4536 
13.4907 

I3-5277 
13-5647 
13.6015 

5-6567 
5-6671 
5-6774 
5-6877 
5.6980 

186 

187 
188 
189 
190 

3  45  96 
3  49  69 
3  53  44 
3  57  21 
3  61  oo 

6  434  856 
6  539  203 
6  644  672 
6  751  269 
6  859  ooo 

13.6382 
13.6748 
13-7113 
13-7477 
13.7840 

5-7083 
5-7185 
5-7287 
5.7388 
5.7489  . 

191 
192 

193 
194 

195 

3  64  81 
3  68  64 
3  7|  49 
3  76  36 
3  80  25 

6  967  871 
7  077  888 
7  189  057 
7  301  384 
7  4H  875 

13.8203 
13.8564 
13-8924 
13.9284 
13.9642 

5.7590 
5.7690 

5.7790 
5.7890 

5-7989 

196 
197 
198 
199 

200 

3  84  16 
3  88  09 
3  92  04 

3  96  OI 

4  oo  oo 

7  529  536 
7  645  373 
7  762  392 
7  880  599 
8  ooo  ooo 

14.0000 

14.0357 
14.0712 
14.1067 
14.1421 

5.8088 
5.8186 
5.8285 

5.8383 
5.8480 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

201 
202 
203 
204 
205 

4  04  oi 
4  08  04 
4  12  09 
4  16  16 
4  20  25 

8  i  20  601 
8  242  408 
8  365  427 
8  489  664 
8  615  125 

14.1774 
14.2127 
14.2478 
14.2829 
14.3178 

5.8578 

5.8675 
5.8771 
5.8868 
5.8964 

206 
207 
208 
209 
2IO 

4  24  36 
4  28  49 
4  32  64 
4  36  81 
4  41  oo 

8  741  816 
8  869  743 
8  998  912 
9  129  329 
9  261  ooo 

14.3527 

14.3875 
14.4222 

14.4568 
14.4914 

5.9059 
5-9I55 
5.9250 

5-9345 

5-9439 

211 
212 
2I3 
214 
215 

4  45  21 
4  49  44 
4  53  69 
4  57  96 
4  62  25 

9  393  93  i 
9  528  128 
9  663  597 
9  800  344 
9  938  375 

14.5258 
14.5602 

14-5945 
14.6287 
14.6629 

5-9533 
5.9627 

5-9721 
5.9814 
5-9907 

216 

217 

218 

2I9 

220 

4  66  56 
4  70  89 
4  75  24 
4  79  61 
4  84  oo 

10  077  696 
10  218  313 
10  360  232 

10  503  459 
10  648  ooo 

14.6969 
14.7309 
14.7648 
14.7986 
14.8324 

6.0000 
6.0092 
6.0185 
6.0277 
6.0368 

221 
222 
223 
224 

225 

488  41 
4  92  84 
4  97  29 
5  oi  76 
5  06  25 

10  793  861 
10  941  048 
ii  089  567 
ii  239  424 
ii  390  625 

14.8661 
14.8997 

H.9332 
14.9666 
1  5.OOOO 

6.0459 
6.0550 
6.0641 
6.0732 
6.0822 

226 

227 
228 
229 

230 

5  10  76 
5  IS  29 
5  19  84 
5  24  4i 
5  29  oo 

ii  543  176 
ii  697  083 
ii  852  352 

12  008  989 
12  167  000 

15.0333 
15.0665 
15.0997 

i5«!327 
15.1658 

6.0912 

6.1002 

6.1091 
6.1180 
6.  1  269 

23I 
232 

233 
234 
235 

5  33  61 
5  38  24 
5  42  89 
5  47  56 
5  52  25 

12  326  391 

12  487  168 
12  649  337 

12  8l2  904 

12  977  875 

15.1987 

15-2315 
15.2643 
15.2971 
15-3297 

6.1358 
6.1446 

6-1534 
6.1622 
6.1710 

236 

III 

239 

240 

55696 
5  61  69 
5  66  44 
5  7i  21 
5  76  oo 

13  144  256 
13  312  053 
13  481  272 
13  651  919 
13  824  ooo 

15.3623 
I5-3948 
15.4272 
15.4596 
15.4919 

6.1797 
6.1885 
6.1972 
6.2058 
6.2145 

241 
242 

243 

244 

245 

5  80  81 
5  85  64 
5  90  49 
5  95  36 
6  oo  25 

13  997  521 
14  172  488 
14  348  907 
14  526  784 
14  706  125 

15.5242 

15-5563 
15-5885 
15.6205 

15-6525 

6.2231 
6.2317 
6.2403 
6.2488 
6.2573 

246 
247 
248 
249 
250 

6  05  16 
6  10  09 
6  15  04 

6  20  01 

6  25  oo 

14  886  936 
15  069  223 
15  252  992 
15  438  249 
15  625  ooo 

15.6844 
15.7162 
15.7480 
15.7797 
15.8114 

6.2658 
6.2743 
6.2828 
6.2912 
6.2996 

Number 

Square. 

Cube.           Square  Root. 

Cube  Root.  ' 

251 
252 

253 
254 
255 

6  30  01 

6  35  04 
6  40  09 
6  45  16 
6  50  25  " 

IS  813  251 
i  6  003  008 
16  194  277 
i  6  387  064 
16  581  375 

15.8430 
15-8745 

15.9060 

15-9374 

15.9687 

6.3080 
6.3164 
6.3247 
6.3330 
6.3413 

256 

257 
258 

259 
260 

6  55  36 
6  60  49 
6  65  64 
6  70  81 
6  76  oo 

16  777  216 

J6  974  593 
17  173  512 

17  373  979 
17  576  ooo 

16.0000 
16.0312 
16.0624 

16.0935 

16.1245 

6.3496 

6-3579 
6.3661 

6.3743 
6.3825 

26l 
262 
263 
264 
265 

6  81  21 
6  86  44 
6  91  69 
6  96  96 

7  02  25 

17  779  581 
17  984  728 
18  191  447 
18  399  744 
18  609  625 

16.1555 
16.1864 

16.2173 
16.2481 

16.2788 

6.3907 
6.3988 
6.4070 
6.4151 
6.4232 

266 
267 
268 
269 
270 

7  07  56 
7  12  89 
7  18  24 
7  23  61 
7  29  oo 

18  821  096 
19  034  163 
19  248  832 
19  465  109 
19  683  ooo 

16.3095 
16.3401 
16.3707 

16.4012 

16.4317 

6.4312 

6.4393 
6.4473 
6.4553 
6.4633 

271 

272 
273 
274 
275 

7  34  4i 
7  39  84 
7  45  29 
7  50  76 
7  56  25 

19  902  511 

20  123  648 
20  346  417 
2O  57O  824 
20  796  875 

16.4621 

16.4924 
16.5227 
16.5529 
16.5831 

6.4713 
6.4792 
6.4872 
6.4951 
6.5030 

276 

277 
278 

279 
280 

7  61  76 
7  67  29 

•  7  72  84 
7  78  41 
7  84  QO 

21  024  576 

21  253  933 

21  484  952 
21  717  639 
21  952  000 

16.6132 

16.6433 
16.6733 
16.7033 
16.7332 

6.5108 
6.5187 
6.5265 

6-5343 
6.5421 

28l 

282 

283 
284 
285 

7  8961 

7  95  24 
8  oo  89 
8  06  56 
8  12  25 

22  188  041 

22  425  768 
22  665  187 
22  906  304 

23  149  I25 

16.7631 

16.7929 
16.8226 
16.8523 
16.8819 

6-5499 
6.5577 
6.5654 

6-5731 
6.5808 

286 
287 
288 
289 
290 

8  17  96 
8  23  69 
8  29  44 
8  35  21 
8  41  oo 

23  393  656 
23  639  903 
23  887  872 

24  137  569 
24  389  ooo 

16.9115 
16.9411 
16.9706 
17.0000 

17.0294 

6.5885 
6.5962 
6.6039 
6.6115 
6.6191 

29I 
292 
293 
294 
295 

8  46  81 
8  52  64 
8  58  49 
8  64  36 
8  70  25 

24  642  171 

24  897  088 

25  !53  757 
25  412  184 

25  672  375 

17.0587 
17.0880 
17.1172 
17.1464 
17.1756 

6.6267 
6.6343 
6.6419 
6.6494 
6.6569 

296 
297 
298 
299 
300 

8  76  16 
8  82  09 
8  88  04 
8  94  01 
9  oo  oo 

25  934  336 
26  198  073 
26  463  592 
26  730  899 
27  ooo  ooo 

17.2047 
17.2337 
17.2627 

17.2916 

17.3205 

6.6644 
6.6719 
6.6794 
6.6869 
6.6943 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

301 
302 
303 
304 
305 

9  06  oi 
9  12  04 
9  18  09 
9  24  16 
9  30  25 

27  270  901 

27  543  608 
27  818  127 
28  094  464 
28  372  625 

17-3494 
17.3781 
17.4069 

I7.4356 
17.4642 

6.7018 
6.7092 
6.7166 

6.7240 

6.73J3 

306 

307 
308 

309 
3IO 

9  36  36 
9  42  49 
9  48  64 
9  54  81 
9  61  oo 

28  652  616 
28  934  443 
29  218  112 
29  503  629 
29  791  ooo 

17.4929 
17.5214 
17.5499 
17.5784 
17.6068 

6.7387 
6.7460 

6-7533 
6.7606 
6.7679 

3" 
312 

3i3 
3H 
3i5 

9  67  21 

9  73  44 
9  79  69 

9  85  96 
9  92  25 

30  080  231 

30  371  328 
30  664  297 
30  959  144 
3i  255  875 

17.6352 
17.6635 
17.6918 

17.7200 
17.7482 

6.7752 
6.7824 
6.7897 
6.7969 
6.8041 

3i6 
3i7 
3i8 
3i9 

320 

998  56 
10  04  89 
10  ii  24 
10  17  61 

10  24  oo 

31  554  496 
31  855  013 
32  157  432 
32  461  759 
32  768  ooo 

17.7764 
17.8045 
17.8326 
17.8606 
17.8885 

6.8113 
6.8185 
6.8256 
6.8328 
6.8399 

321 
322 
323 
324 
325 

10  30  41 
10  36  84 
10  43  29 
10  49  76 
10  56  25 

33  076  161 
33  386  248 
33  698  267 
34  012  224 
34  328  125 

17.9165 
17.9444 
17.9722 
18.0000 
18.0278 

6.8470 
6.8541 
6.8612 
6.8683 
6.8753 

326 
327 
328 
329 
330 

10  62  76 
10  69  29 
10  75  84 
10  82  41 
10  89  oo 

34  645  976 
34  965  783 
35  287  552 
35  611  289 
35  937  ooo 

18.0555 
18.0831 

18.1108 
18.1384 
18.1659 

6.8824 
6.8894 
6.8964 
6.9034 
6.9104 

331 
332 
333 
334 
335 

10  95  61 
ii  02  24 
ii  08  89 
ii  15  56 

II  22  25 

36  264  691 
36  594  368 
36  926  037 
37  259  704 
37  595  375 

18.1934 
18.2209 
18.2483 
18.2757 
18.3030 

6.9174 
6.9244 

6-93  i  3 
6.9382 
6.9451 

336 
337 
338 
339 
340 

II  28  96 

ii  35  69 
ii  42  44 
ii  49  21 
ii  56  oo 

37  933  056 
38  272  753 
38  614  472 
38  958  219 
39  304  ooo 

18.3303 
18.3576 
18.3848 
18.4120 
18.4391 

6.9521 
6.9589 
6.9658 
6.9727 

6.9795 

34i 
342 
343 
344 
345 

ii  62  81 
ii  69  64 
ii  76  49 
ii  83  36 
ii  90  25 

39  651  821 
40  ooi  688 

40  353  6°7 
40  707  584 
41  063  625 

18.4662 

18.4932 
18.5203 
18.5472 
18.5742 

6.9864 
6.9932 
7.0000 
7.0068 
7.0136 

346 
347 
348 
349 
350 

ii  97  16 
12  04  09 

12  II  04 
12  l8  OI 

12  25  00 

41  421  736 
-41  781  923 
42  144  192 
42  508  549 
42  875  ooo 

18.6011 
18.6279 
18.6548 
18.6815 
18.7083 

7.0203 
7.0271 

7.0338 
7.0406 

7-0473 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

351 
352 

353 

354 
355 

12  32  OI 

12  39  04 

12  46  09 

12  53  16 

12  60  25 

43  243  55i 
43  614  208 
43  986  977 
44  361  864 
44  738  875 

18.7350 
18.7617 
18.7883 
18.8149 
18.8414 

7.0540 
7.0607 
7.0674 
7.0740 
7.0807 

356 

357 
358 
359 
360 

12  67  36 

12  74  49 

12  8  1  64 

12  88  81 

12  96  00 

45  118  016 
45  499  293 
45  882  712 
46  268  279 
46  656  ooo 

18.8680 
18.8944 
18.9209 
18.9473 
18.9737 

7.0873 
7.0940 
7.1006 
7.1072 
7.1138 

361 
362 
363 
364 
365 

13  03  21 

13  10  44 

13  17  69 
13  24  96 
13  32  25 

47  045  88  i 
47  437  928 
47  832  147 
48  228  544 
48  627  125 

19.0000 
19.0263 
19.0526 
19.0788 
19.1050 

7.1204 
7.1269 

7-1335 
7.1400 
7.1466 

366 

367 
368 

369 

370 

13  39  56 
13  46  89 

13  54  24 
13  61  61 
13  69  oo 

49  027  896 
49  430  863 
49  836  032 
50  243  409 
50  653  ooo 

I9.I3II 
19.1572 

19-  '833 
19.2094 

I9-2354 

7.I53I 

7'1£6 
7.l66l 

7.1726 
7.I79I 

37i 
372 
373 
374 

375 

13  76  41 
13  83  84 
13  91  29 
13  98  76 
14  06  25 

51  064  811 
51  478  848 
51  895  117 
52  313  624 

52  734  375 

19.2614 
19.2873 
19.3132 

I9-339I 
19.3649 

7.1855 
7.1920 

7.1984 
7.2048 
7.2II2 

376 
377 
378 

379 

380 

H  13  76 

14  21  29 
14  28  84 
14  36  41 

14  44  oo 

53  157  376 
53  582  633 
54  oio  152 

54  439  939 
54  872  ooo 

19.3907 
19.4165 
19.4422 
19.4679 
19.4936 

7.2177 
7.2240 

7.2304 
7.2368 
7.2432 

38i 
382 

383 
384 
385 

14  51  61 
14  59  24 
14  66  89 
14  74  56 

14  82  25 

55  306  34i 
55  742  968 
56  181  887 
56  623  104 
57  066  625 

19.5192 
19.5448 
19.5704 

1  9-  59  59 
19.6214 

7-2495 
7.2558 
7.2622 
7-2685 
7.2748 

386 

387 
388 
389 
390 

14  89  96 
14  97  69 
15  05  44 
15  13  21 

IS  21  OO 

57  512  456 
57  960  603 
58  411  072 
58  863  869 
59  319  ooo 

19.6469 
19.6723 
19.6977 
19.7231 
19.7484 

7.28II 

7.2874 
7.2936 
7.2999 

7.3061 

39i 
392 
393 
394 
395 

15  28  81 

15  36  64 
15  44  49 

IS  52  36 
15  60  25 

59  776  471 
60  236  288 
60  698  457 
61  162  984 
61  629  875 

19-7737 
19.7990 
19.8242 
19.8494 
19.8746 

7-3I24 
7.3186 
7.3248 
7.3310 
7-3372 

396 
397 
398 
399 
400 

15  68  16 
15  76  09 
15  84  04 
15  92  01 
16  oo  oo 

62  099  136 
62  570  773 
63  044  792 
63  521  199 

64  ooo  ooo 

19.8997 
19.9249 
19.9499 
19.9750 
20.0000 

7-3434 
7.3496 
7.3558 
7.3619 
7.3681 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

401 
4O2 

403 
404 

405 

16  08  01 
16  16  04 
16  24  09 
16  32  16 
16  40  25 

64  481  2OI 
64  964  808 
65  450  827 

65  939  264 
66  430  125 

20.0250 
20.0499 
20.0749 
20.0998 
20.1246 

7-3742 
7.3803 
7.3864 

7.3925 
7.3986 

406 
407 
408 
409 
410 

16  48  36 
16  56  49 
16  64  64 
16  72  81 
16  81  oo 

66  923  416 
67  419  143 
67  917  312 
68  417  929 
68  921  ooo 

20.1494 
20.1742 
20.1990 
20.2237 
20.2485 

7.4047 
7.4108 
7.4169 
7.4229 
7.4290 

411 
413 

413 
414 

415 

16  89  21 
16  97  44 
17  05  69 
17  13  96 

17  22  25 

69  426  531 
69  934  528 
70  444  997 
70  957  944 
7i  473  375 

20.2731 
20.2978 
20.3224 
20.3470 
20.3715 

7-435° 
7.4410 
7.4470 
7.4530 

7-459° 

416 

417 
4l8 
419 

420 

I7  30  56 

17  38  89 
17  47  24 
17  55  61 

17  64  oo 

71  991  296 
72  5"  713 
73  °34  632 
73  56o  059 
74  088  ooo 

20.3961 
20.4206 
20.4450 
20.4695 
20.4939 

7.4650 
7.4710 

7-477° 
7.4829 
7.4889 

42I 
422 

423 
424 

425 

17  72  41 
17  80  84 
17  89  29 

17  97  76 

18  06  25 

74  618  461 
75  151  448 
75  686  967 
76  225  024 
76  765  625 

20.5183 
20.5426 
20.5670 
20.5913 
20.6155 

7.4948 
7.5007 
7.5067 
7.5126 
7-5185 

426 

427 

428 

429 

43° 

18  14  76 
18  23  29 
18  31  84 
18  40  41 
18  49  oo 

77  3°8  776 
77  854  483 
78  402  752 

78  953  589 
79  507  ooo 

20.6398 
20.6640 
20.6882 
20.7123 
20.7364 

7.5244 
7.53°2 
7.5301 
7.5420 

7.5478 

43i 
432 
433 
434 
435 

18  57  61 
18  66  24 
18  74  89 
18  83  56 
18  92  25 

80  062  991 
80  621  568 
81  182  737 
8  i  746  504 
82  312  875 

20.7605 
20.7846 
20.8087 
20.8327 
20.8567 

7-5537 
7-5595 
7.5654 
7.5712 
7-577° 

436 

438 
439 

440 

19  oo  96 
19  09  69 
19  18  44 
19  27  21 
19  36  oo 

82  881  856 

83  453  453 
84  027  672 
84  604  519 
85  184  ooo 

20.8806 
20.9045 
20.9284 
20.9523 
20.9762 

7-5828 
7.5886 

7-5944 
7.6001 
7-6059 

441 

442 
443 
444 
445 

19  44  81 

19  53  64 
19  62  49 
19  71  36 
19  80  25 

85  766  121 

86  350  888 
86  938  307 
87  528  384 
88  121  125 

21.0000 
21.0238 
21.0476 
21.0713 
21.0950 

7.6117 
7.6174 
7.6232 
7.6289 
7.6346 

446 

447 
448 

449 
45° 

19  89  16 
1998  09 
20  07  04 
20  16  01 

20  25  00 

88  716  536 
89  314  623 

89  9i5  392 
90  518  849 
91  125  ooo 

2I.II87 
21.1424 
21.1000 
21.1896 
21.2132 

7.6403 
7.6460 
7.6517 
7.6574 
7-6631 

Numbe 

Square. 

Cube.           Square  Root. 

Cube  Root. 

451 
452 

453 
454 
455 

20  34  01 
20  43  04 

20  52  09 

20  61  16 

20  70  25 

91  733  851 
92  345  408 
92  959  677 
93  576  664 
94  i96  375 

21.2368 
21.2603 
21.2838 
21.3073 
21.3307 

7.6688 
7.6744 
7.6801 
7.6857 
7.6914 

456 

457 
458 
459 
460 

20  79  36 
20  88  49 
20  97  64 
21  06  81 

21  l6  00 

94  818  816 

95  443  993 
96  071  912 
96  702  579 
97  336  ooo 

21.3542 
21.3776 
21.4009 
21.4243 
21.4476 

7.6970 
7.7026 
7.7082 
7.7138 
7-7I94 

461 
462 

463 
464 
465 

21  25  21 

21  34  44 
21  43  69 

21  52  96 
21  62  25 

97  972  181 
98  611  128 
99  252  847 

99  897  344 
100  544  625 

21.4709 
21.4942 
21.5174 
21.5407 
21.5639 

7.7250 
7.7306 
7.7362 
7.7418 
7-7473 

466 
467 
468 
469 

470 

21  71  56 
21  80  89 
21  90  24 

21  99  61 

22  09  00 

101  194  696 
101  847  563 

102  503  232 

103  161  709 

103  823  ooo 

21.5870 
21.6102 

21.6333 

21.6564 

21.6795 

7.7529 
7.7584 
7.7639 

7-7695 
7.7750 

47i 
472 

473 
474 
475 

22  l8  41 
22  27  84 

22  37  39 

22  46  76 
22  56  25 

104  487  III 

105  154  048 
105  823  817 
I  06  496  424 
107  171  875 

21.7025 
21.7256 

21.7486 

21.7715 
21.7945 

7-7805 
7.7860 

7.7915 
7.7970 
7.8025 

476 

477 
478 

479 
480 

22  65  76 

22  75  29 

22  84  84 

22  94  41 
23  04  oo 

107  850  176 

i°8  53i  333 
109  215  352 
109  902  239 
no  592  ooo 

21.8174 
21.8403 
21.8632 
21.8861 

21.9089 

7.8079 

7.8134 
7.8l88 
7.8243 
7.8297 

481 
482 
483 
484 
485 

23  13  61 

23  23  24 
23  32  89 
23  42  56 
23  52  25 

in  284  641 
in  980  168 
112  678  587 

"3  379  904 
114  084  125 

21.9317 
21.9545 
21.9773 

22.0000 
22.0227 

7.8352 
7.8406 
7.8460 
7.8514 
7.8568 

486 

487 
488 
489 
490 

23  61  96 
23  71  69 
23  81  44 
23  91  21 
24  01  oo 

114  791  256 
115  501  303 
116  214  272 
116  930  169 
117  649  ooo 

22.0454 
22.0681 
22.0907 
22.1133 
22.1359 

7.8622 
7.8676 
7.8730 
7.8784 
7.8837 

491 
492 

493 
494 
495 

24  10  81 

24  20  64 

24  3°  49 
24  40  36 
24  50  25 

118  370  771 
119  095  488 
119  823  157 
120  553  784 
121  287  375 

22.1585 

22.1811 

22.2036 
22.2261 
22.2486 

7.8891 

7.8944 
7.8998 
7.9051 

7-9I05 

496 
497 
498 
499 
500 

24  60  16 
24  70  09 
24  80  04 
24  90  01 
25  oo  oo 

122  023  936 
122  763  473 

"3  5°5  992 
124  251  499 
125  ooo  ooo 

22.2711 

22.2935 
22.3159 
22.3383 
22.3607 

7.9158 
7.9211 
7.9264 

7-93  i  7 
7.9370 

'Number     Square. 

Cube. 

Square  Root. 

Cube  Root. 

501 

S02 

5°3 
5°4 
5°5 

25  10  01 
25  20  04 
25  30  09 
25  40  16 
25  50  25 

125  751  501 
126  506  008 
127  263  527 
128  024  064 
128  787  625 

22.3830 
22.4054 
22.4277 
22.4499 
22.4722 

7.9423 
7.9476 

7-95-8 
7-958i 
7-9634 

506 

507 
508 

5°9 
Sio 

25  60  36 
25  70  49 
25  80  63 
25  90  81 
26  01  oo 

129  554  216 
130  323  843 
131  096  512 
131  872  229 
132  651  ooo 

22.4944 
22.5167 
22.5389 
22.5610 
22.5832 

7.9686 

7-9739 
7.9791 

7.9843 
7.9896 

5" 
512 

5i3 

5H 
5i5 

26  II  21 

26  21  44 
26  31  69 
26  41  96 
26  52  25 

133  432  831 
134  217  728 
135  005  697 

J35  796  744 
136  590  875 

22.6053 
22.6274 
22.6495 
22.6716 
22.6936 

7.9948 
8.0000 
8.0052 
8.0104 
8.0156 

5i6 
5i7 
518 

5i9 

520 

26  62  56 
26  72  89 
26  83  24 
26  93  61 
27  04  oo 

137  388  096 
138  188  413 
138  991  832 

'39  798  359 
140  608  ooo 

22.7156 
22.7376 
22.7596 
22.7816 
22.8035 

8.0208 
8.0260 
8.0311 
8.0363 
8.0415 

521 
522 

523 
524 

525 

27  14  41 
27  24  84 
27  35  29 
27  45  76 
27  56  25 

141  420  761 
142  236  648 
143  055  667 
143  877  824 
144  703  125 

22.8254 
22.8473 
22.8692 
22.8910 
22.9129 

8.0466 
8.0517 
8.0569 
8.0620 
8.0671 

526 

527 
528 

529 

530 

27  66  76 
27  77  29 
27  87  84 
27  98  41 
28  09  oo 

M5  53i  576 
146  363  183 
147  197  952 
148  035  889 
148  877  ooo 

22.9347 
22.9565 
22.9783 
23.0000 
23.0217 

8.0723 
8.0774 
8.0825 
8.0876 
8.0927 

53i 

532 
'  533 
534 
535 

28  19  61 
28  30  24 
28  40  89 
28  51  56 
28  62  25 

149  721  291 
150  568  768 

I5I  4i9  437 
152  273  304 

153  13°  375 

23-0434 

23.0868 
23.1084 
23.1301 

8.0978 
8.1028 
8.1079 
8.1130 
8.1  180 

536 

s 

539 
540 

28  72  96 
28  83  69 
28  94  44 

29  05  21 

29  16  oo 

i53  990  656 
154  854  153 
155  720  872 
156  590  819 
157  464  ooo 

23.1517 

23.1733 
23.1948 
23.2164 
23.2379 

8.1231 
8.1281 
8.1332 
8.1382 

8-1433 

54i 
542 
543 
544 
545  1 
546 
547 
548 
549 
550 

29  26  81 
29  37  64 
29  48  49 

29  59  36 
29  70  25 

158  340  421 

159  220  088 

160  103  007 

I  60  989  184 

161  878  625 

23.2594 
23.2809 
23.3024 
23.3238 
23.3452 

8.1483 

8-1533 
8.  1  583 
8-1633 
8.1683 

29  81  16 
29  92  09 
30  03  04 
30  14  01 
30  25  oo 

162  771  336 
163  667  323 

164  566  592 

165  469  149 

i  66  375  ooo 

23.3666 
23.3880 
23.4094 
23.4307 
23.4521 

8-1733 
8.1783 
8.1833 
8.1882 
8.1932 

Numbe 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

551 
55* 

553 

554 
555 

30  36  o  i 
3°  47  04 

30  58  09 

30  69  16 

30  80  25 

167  284  151 

i  68  196  608 
169  112  377 
170  031  464 
170  953  875 

23-4734 
23.4947 
23.5160 
23-5372 
23-5584 

8.1982 
8.2031 

8.  208  1 
8.2130 
8.2180 

556 

III 
1 

30  91  36 
31  02  49 

31  '3  64 
31  24  81 
31  36  oo 

171  879  616 
172  808  693 
173  741  112 
174  676  879 
175  616  ooo 

23-5797 
23.6008 
23.6220 
23.6432 
23.6643 

8.2229 
8.2278 
8.2327 

8.2377 
8.2426 

561 
562 

563 
564 

565 

31  47  21 

31  58  44 
31  69  69 
31  80  96 
31  92  25 

176  558  481 

177  504  328 

178  453  547 
179  406  144 
180  362  125 

23-6854 
23.7065 
23.7276 
23.7487 
23.7697 

8.2475 
8.2524 
8.2573 
8.2621 
8.2670 

566 

567 
568 

569 

570 

•  32  03  56 
32  14  89 
32  26  24 
32  37  61 
32  49  oo 

181  321  496 
182  284  263 
183  250  432 
184  220  009 
185  193  ooo 

23.7908 
23.8118 
23.8328 

23.8537 
23.8747 

8.2719 
8.2768 
8.2816 
8.2865 
8.2913 

57i 
572 
573 
574 

575 

32  60  41 
32  71  84 
32  83  29 
32  94  76 
33  06  25 

186  169  411 
187  149  248 
188  132  517 
189  119  224 
190  109  375 

23.8956 
23.9165 
23.9374 
23-9583 
23.9792 

8.2962 
8.3010 
8.3059 
8.3107 
8.3155 

576 
577 
578 

579 
580 

33  17  76 
33  29  29 
33  40  84 
33  52  41 
33  64  op 

191  IO2  976 

192  100  033 

*93  I0°  SS2 
194  104  539 

195  112  OOO 

24.OOOO 
24.0208 
24.0416 
24.0624 
24.0832 

8.3203 
8.3251 
8.3300 
8.3348 
8.3396 

58i 
582 

583 
584 
585 

33  75  61 

33  87  24 
33  98  89 
34  10  56 
34  22  25 

196  122  941 
197  137  368 
I98  155  287 
199  176  704 
200  201  625 

24.1039 
24.1247 
24.1454 
24.1661 
24.1868 

8-3443 
8.3491 

8-3539 
8.3587 
8.3634 

586 

587 
588 
589 
590 

34  33  96 
34  45  69 
34  57  44 
34  69  21 
34  8  i  oo 

2O  I  230  056 
2O2  262  003 
203  297  472 
204  336  469 

205  379  ooo 

24.2074 
24.2281 
24.2487 
24.2693 
24.2899 

8.3682 
8-3730 
8-3777 
8.3825 
8.3872 

59i 
592 
593 
594 
595 

34  92  81 
35  04  64 

35  l6  49 
35  28  36 
35  40  25 

206  425  071 
207  474  688 
208  527  857 
209  584  584 

210  644  875 

24.3105 

24-33" 
24.3516 
24.3721 
24.3926 

8.3919 

8.3967 
8.4014 
8.4061 
8.4108 

596 
597 
598 
599 
600 

35  52  16 
35  64  09 
35  76  04 
35  88  01 
36  oo  oo 

211  708  736 
212  776  173 
213  847  192 

214  921  799 
216  ooo  ooo 

24.4131 

24-4336 
24.4540 

24-4745 
24.4949 

8.4155 
8.4202 
8.4249 
8.4296 
8.4343 

Number     Square.             Cube. 

Square  Root. 

Cube  Root. 

60  1 
602 
603 
604 
605 
606 
607 
608 
609 

610 

36  12  OI 
36  24  04 
36  36  09 

36  48  16 
36  60  25 

217  081  801 
218  167  208 
219  256  227 

220  348  864 

221  445  125 

24-5153 
24-5357 
24.5561 
24.5764 
24.5967 

8.4390 

8.4437 

8.4484 
8.4530 
8-4577 

36  72  36 
36  84  49 
36  96  64 
37  08  81 
37  21  oo 

222  545  016 
223  648  543 
224  755  712 
225  866  529 
226  981  ooo 

24.6171 
24.6374 
24.6577 
24.6779 
24.6982 

8.4623 
8.4670 
8.4716 
8.4763 
8.4809 

in 

612 

613 
6i4 
615 

37  33  21 
37  45  44 
37  57  69 
37  69  96 
37  82  25 

228  099  131 

229  220  928 

230  346  397 
231  475  544 
232  608  375 

24.7184 
24.7386 
24.7588 
24.7790 
24.7992 

8.4856 
8.4902 
8.4948 
8.4994 
8.5040 

616 
617 
618 
619 

620 

37  94  56 
38  06  89 
38  19  24 
38  31  61 
38  44  oo 

233  744  896 
234  885  113 
236  029  032 
237  176  659 
238  328  ooo 

24.8193 

24-8395 
24.8596 
24.8797 
24.8998 

8.5086 
8.5132 
8.5178 
8.5224 
8.5270 

621 
622 
623 
624 
625 

38  56  41 
38  68  84 
38  81  29 

38  93  76 
39  06  25 

239  483  06  I 
240  641  848 
241  804  367 
242  970  624 
244  140  625 

24.9199 

24.9399 
24.9600 
24.9800 
25.OOOO 

8-53'b 
8.5362 
8.5408 
8-5453 
8.5499 

626 
627 

628 
629 
630 

39  l8  76 
39  3i  29 
39  43  84 
39  56  4i 
39  69  oo 

245  3'4  376 
246  491  883 
247  673  152 
248  858  189 
250  047  ooo 

25.0200 
25.0400 
25.0599 
25.0799 
25.0998 

8-5544 
8.5590 

8.5635 
8.5681 
8.5726 

631 
632 
633 
634 
635 

39  8  i  61 
39  94  24 
40  06  89 
40  19  56 
40  32  25 

251  239  591 
252  435  968 
253  636  137 
254  840  104 
256  047  875 

25.1197 
25.1396 

25.1595 
25.1794 
25.1992 

8.5772 
8.5817 
8.5862 
8.5907 
8.5952 

636 

637 
638 

639 
640 

40  44  96 
40  57  69 
40  70  44 

40  83  21 

40  96  oo 

257  259  456 
258  474  853 
259  694  072 
260  917  119 
262  144  ooo 

25.2190 
25.2389 
25.2587 
25.2784 
25.2982 

8-5997 
8.6043 
8.6088 
8.6132 
8.6177 

641 
642 

643 
644 

645 

41  08  81 

41  21  64 

4i  34  49 
4i  47  36 
41  60  25 

263  374  721 
264  609  288 
265  847  707 
267  089  984 
268  336  125 

25.3180 
25.3377 
25-3574 
25.3772 
25.3969 

8.6222 
8.6267 
8.6312 
8.6357 
8.6401 

646 
647 
648 
649 
650 

41  73  16 
41  86  09 
41  99  04 
42  12  01 
42  25  oo 

269  586  136 
270  840  023 
272  097  792 

273  359  549 
274  625  ooo 

25.4165 
25.4362 
25-4558 
25-4755 
25.49$! 

8.6446 
8.6490 
8.6535 
8.6579 
8.6624 

NumberJ     Square. 

Cube. 

Square  Root. 

Cube  Root. 

651 

652 

653 
654 
655 

42  38  01 
42  51  04 
42  64  09 
42  77  16 
42  90  25 

275  894  451 
277  167  808 
278  445  077 
279  726  264 
281  on  375 

25.5H7 
25.5343 
25.5539 
25-5734 
25.5930 

8.6668 

8.6713 
8.6757 
8.6801 
8.6845 

656 

657 

658 

659 
660 

43  03  36 
43  l6  49 
43  29  64 
43  42  81 
43  56  oo 

282  300  416 

283  593  393 
284  890  312 
286  191  179 
287  496  ooo 

25.6125 
25.6320 
25.6515 
25.6710 
25.6905 

8.6890 
8.6934 
8.6978 

8.7022 

8.7066 

66  1 
662 
663 
664 
665 

43  69  21 

43  82  44 
43  95  69 
44  08  96 
44  22  25 

288  804  781 

290  117  528 
291  434  247 
292  754  944 
294  079  625 

25.7099 
25.7294 
25.7488 
25.7682 
25.7876 

8.7110 

8.7154 
8.7198 
8.7241 
8.7285 

666 
667 
668 
669 
670 

44  35  56 
44  48  89 
44  62  24 

44  75  61 
44  89  oo 

295  408  296 
296  740  963 
298  077  632 
299  4i8  309 
300  763  ooo 

25.8070 
25.8263 
25.8457 
25.8650 
25.8844 

8.7329 

8.7373 
8.7416 
8.7460 
8.7503 

671 
672 

673 
674 

675 

45  °2  4i 
45  !5  84 
45  29  29 
45  42  76 
45  56  25 

302  III  711 

303  464  448 

304  821  217 
306  182  024 

307  546  875 

25-9037 
25.9230 
25.9422 
25.9615 
25.9808 

8.7547 
8.7590 
8.7634 

8.7677 
8.7721 

676 

677 
678 
679 
680 

45  69  76 
45  83  29 
45  96  84 
46  10  41 
46  24  oo 

308  915  776 

310  288  733 
311  665  752 
313  046  839 
314  432  ooo 

26.OOOO 
26.0192 
26.0384 
26.0576 
26.0768 

8.7764 
8.7807 
8.7850 
8.7893 
8.7937 

68  1 
682 
683 
684 
685 

46  37  61 
46  51  24 
46  64  89 
46  78  56 
46  92  25 

315  821  241 
317  214  568 
318  611  987 
320  013  504 
321  419  125 

26.0960 
26.1151 
26.1343 
26.1534 
26.1725 

8.7980 
8.8023 
8.8066 
8.8109 
8.8152 

686 
687 
688 
689 
690 

47  05  96 
47  19  69 
47  33  44 
47  47  21 
47  61  oo 

322  828  856 
324  242  703 
325  660  672 
327  082  769 
328  509  ooo 

26.1916 
26.2IO7 
26.2298 
26.2488 
26.2679 

8.8194 
8.8237 
8.8280 
8.8323 
8.8366 

691 
692 

693 
694 
695 

47  74  81 
47  88  64 
48  02  49 
48  16  36 
48  30  25 

329  939  371 
33i  373  888 
332  812  557 
334  255  384 
335  702  375 

26.2869 
26.3059 
26.3249 

26.3439 
26.3629 

8.8408 
8.8451 

8.8493 
8.8536 
8.8578 

696 

697 
698 

699 

700 

48  44  16 
48  58  09 
48  72  04 
48  86  01 
49  oo  oo 

337  153  536 
338  608  873 
340  068  392 
341  532  099 
343  ooo  ooo 

26.3818 
26.4008 
26.4197 
26.4386 
26.4575 

8.8621 
8.8663 
8.8706 
8.8748 
8.8790 

Numbe 

Square. 

Cube. 

Square  Root. 

Cube  Roor. 

701 
702 

703 
704 

70S 

49  14  oi 
49  28  04 
49  42  09 

49  56  l6 
49  70  25 

344  472  101 
345  948  408 
347  428  927 
348  913  664 
350  402  625 

26.4764 

26.4953 
26.5141 
26.5330 
26.5518 

8.8833 

8.8875 
8.89.7 
8.8959 
8.9001 

706 
707 
708 
709 
710 

49  84  36 
49  98  49 

50  12  64 

50  26  81 
50  41  oo 

35i  895  816 

353  393  243 
354  894  912 
356  400  829 
357  911  ooo 

26.5707 
26.5895 
26.6083 
26.6271 
26.6458 

8.9043 
8.9085 
8.9127 
8.9(69 
8.9211 

711 
712 
713 
7M 
715 

So  55  21 
5°  69  44 
50  83  69 
50  97  96 
51  !2  25 

359  425  43i 
360  944  128 
362  467  097 

363  994  344 
365  525  875 

26.6646 
26.6833 
26.7021 
26.7208 
26.7395 

8.9253 
8.9295 

8-9337 
8.9378 
8.9420 

716 
717 
718 
719 

720 

51  26  56 
51  40  89 

51  I5  *4 
51  69  61 

51  84  oo 

367  06  i  696 
368  601  813 
370  146  232 
371  694  959 

373  248  ooo 

26.7582 
26.7769 
26.7955 
26.8142 
26.8328 

8.9462 
8.9503 
8-9545 
8.9587 
8.9628 

72I 
722 

723 
724 

725 

51  98  41 

52  12  84 

52  27  29 
52  41  76 
52  56  25 

374  805  361 
376  367  048 
377  933  °67 
379  5°3  424 
381  078  125 

26.8514 
26.8701 
26.8887 
26.9072 
26.9258 

8.9670 
8.9711 

8.9752 
8.9794 

89835 

726 
727 
728 
729 
730 

52  70  76 
52  85  29 

52  99  84 

53  H  4i 
53  29  GO" 

382  657  176 
384  240  583 
385  828  352 
387  420  489 
389  017  ooo 

26.9444 
26.9629 
26.9815 

27.OOOO 
27.0185 

8.9876 
8.9918 

8.9959 
9.0000 
9.0041 

731 

732 

733 
734 
735 

53  43  61 
53  58  24 
53  72  89 
53  87  56 
54  02  25 

390  617  891 

392  223  168 
393  832  837 
395  446  904 
397  065  375 

27.0370 
27.0555 
27.0740 
27.0924 
27.1109 

9.0082 
9.0123 
9.0164 
9.0205 
9.0246 

736 
737 
738 
739 
740 

54  16  96 
54  31  69 
54  46  44 
54  61  21 
54  76  oo 

398  688  256 
400  315  553 
401  947  272 

403  583  419 
405  224  ooo 

27.1293 
27.1477 
27.1662 
27.1846 
27.2029 

9.0287 
9.0328 
9.0369 
9.0410 

9.0450 

74i 
742 

743 
744 
745 

54  9°  81 
55  °5  64 
55  20  49 
55  35  36 
55  5°  25 

406  869  021 
408  518  488 
410  172  407 
411  830  784 

413  493  625 

27.2213 
27.2397 
27.2580 
27.2764 
27.2947 

9.0491 
9.0532 
9.0572 
9.0613 
9.0654 

746 

747 
748 

749 

750 

55  65  16 
55  80  09 

55  95  04 
56  10  oi 
56  25  oo 

415  160  936 
416  832  723 
418  508  992 
420  189  749 
421  875  ooo 

27.3130 

27.3313 
27.3496 
27.3679 
27.3861 

9.0694 

9-0735 
9.0775 
9.0816 
9.08  s6 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

751 
752 

753 
754 

755 

56  40  oi 

56  55  04 
56  70  09 
56  85  16 
57  oo  25 

423  564  751 

425  259  008 

426  957  777 
428  66  i  064 
430  368  875 

27.4044  . 
27.4226 
27.4408 
27.4591 
27-4773 

9.0896 

9.0937 
9.0977 
9.IOI7 

9-Jo57 

756 
757 
758 
759 
760 

57  15  3^ 
57  30  49 
57  45  64 
57  60  81 
57  76  oo 

432  081  216 
433  798  093 
435  5i9  512 
437  245  479 
438  976  ooo 

27.4955 
27.5136 
27-53I8 
27-5500 
27.5681 

9.1098 
9.1138 
9.1178 
9.1218 
9.1258 

761 
762 

763 
764 

765 

57  91  21 
58  06  44 
58  21  69 
58  36  96 
58  52  25 

440  711  08  i 
442  450  728 
444  194  947 

445  943  744 
447  697  125 

27.5862 
27.6043 
27.6225 
27.6405 
27.6586 

9.1298 
9-1338 
9.1378 
9.1418 
9.1458 

766 
767 
768 
769 
770 

58  67  56 
58  82  89 
58  98  24 

59  13  6l 
59  29  oo 

449  455  096 
451  217  663 
452  984  832 

454  756  609 
456  533  ooo 

27.6767 
27.6948 
27.7128 
27.7308 
27.7489 

9.1498 
9-1537 
9-*577 
9.1617 

9-^57 

771 
772 
773 
774 
775 

59  44  41 
59  59  84 
59  75  29 
59  90  76 
60  06  25 

458  314  on 
460  099  648 
461  889  917 
463  684  824 
465  484  375 

27.7669 
27.7849 
27.8029 
27.8209 
27.8388 

9.1696 
9.1736 

9.1775 
9.1815 
9.1855 

776 

777 
778 

779 
780 

60  21  76 

60  37  29 
60  52  84 
60  68  41 
60  84  oo 

467  288  576 
469  097  433 
470  910  952 
472  729  139 
474  552  ooo 

27.8568 
27.8747 
27.8927 
27.9106 
27.9285 

9.1894 
9-1933 
9.1973 
9.2012 
9.2052 

78i 
782 

783 
784 
785 

60  99  61 
6  1  15  24 
61  30  89 
61  46  56 
61  62  25 

476  379  54i 
478  211  768 
480  048  687 
481  890  304 
483  736  625 

27.9464 
27.9643 
27.9821 
28.OOOO 
28.0179 

9.2091 
9.2130 
9.2170 
0.2209 
9.2248 

786 

787 
788 
789 
790 

61  77  96 
61  93  69 
62  09  44 

62  25  21 

62  41  oo 

485  587  656 
487  443  403 
489  3°3  872 
491  169  069 
493  °39  ooo 

28.0357 
28.0535 
28.0713 
28.0891 
28.1069 

9.2287 
9.2326 
9.2365 
9.2404 
9.2443 

791 
792 
793 
794 
795 

62  56  81 
62  72  64 
62  88  49 
63  04  36 
63  20  25 

494  9*3  671 
496  793  088 
498  677  257 
500  566  184 
502  459  875 

28.1247 
28.1425 
28.1603 
28.1780 
28.1957 

9.2482 
9.2521 
9.2560 
9.2599 
9.2638 

796 

797 
798 

799 

800 

63  36  16 
63  52  09 
63  68  04 
63  84  oi 
64  oo  oo 

504  358  336 
506  261  573 
508  169  592 
510  082  399 
512  ooo  ooo 

28.2135 
28.2312 
28.2489 
28.2666 
28.2843 

9.2677 
9.2716 

9-2754 
9-2793 
9.2832 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root 

80  1 
802 
803 
804 
805 

64  i  6  01 

64  32  04 

64  48  09 
64  64  16 
64  80  25 

513  922  401 
515  849  608 

517  ?8i  627 

519  718  464 
521  660  125 

28.3019 
28.3196 
28-3373 
28.3549 
28.3725 

9.2870 
9.2909 
9.2948 
9.2986 
9-3025 

Sob 
807 
808 
809 
810 

64  96  36 
65  12  49 
65  28  64 
65  44  81 
65  61  oo 

523  606  616 

525  557  943 
527  514  112 
529  475  129 
531  441  ooo 

28.3901 
28.4077 
28.4253 
28.4429 
28.4605 

9.3063 
9.3102 
9.3140 

9-3  '79 
9.3217 

811 
812 

8,'3 
Si  4 

815 

65  77  21 

65  93  44 
66  09  69 
66  25  96 
66  42  25 

533  4"  73» 
535  387  328 
537  367  797 
539  353  M4 
54i  343  375 

28.478l 
28.4956 
28.5132 
28.5307 
28.5482 

9-3255 
9.3294 

9-3332 
9-3370 
9.3408 

816 
817 
818 
819 
820 

66  58  56 
66  74  89 
66  91  24 
67  07  61 
67  24  oo 

543  338  496 
545  338  S13 
547  343  432 
549  353  259 
551  368  ooo 

28.5657 
28.5832 
28.6007 
28.6l82 
28.6356 

9-3447 
9-3485 
9-3523 
9-356i 
9-3599 

821 
822 
823 
824 
825 

67  40  41 
67  56  84 
67  73  29 
67  89  76 
68  06  25 

553  387  66  i 
555  412  248 
557  441  767 
559  476  224 
561  515  625 

28.6531 
28.6705 
28.6880 
28.7054 
28.7228 

9-3637 
9-3675 
9-3713 
9-3751 
9-3789 

826 
827 
828 
829 
830 

68  22  76 
68  39  29 
68  55  84 
68  72  41 
68  89  oo 

563  559  976 
565  609  283 
567  ^63  552 
569  722  789 
571  787  ooo 

28.7402 
28.7576 
28.7750 
28.7924 
28.8097 

9.3827 

9-3865 
9.3902 
9.3940 
9-3978 

831 
832 
833 
834 
835 

69  05  61 

69  22  24 
69  38  89 

69  55  56 
69  72  25 

573  856  191 
575  930  368 
578  009  537 
580  093  704 
582  182  875 

28.8271 
28.8444 
28.8617 
28.8791 
28.8964 

9.4016 

9-4053 
9.4091 
9.4129 
9.4166 

836 

837 
838 

839 
840 

69  88  96 
70  05  69 
70  22  44 
70  39  21 
70  56  oo 

584  277  056 
586  376  253 
588  480  472 
590  589  719 
592  704  ooo 

28.9137 
28.9310 
28.9482 
28.9655 
28.9828 

9.4204 
9.4241 

9.4279 
9.4316 

9-4354 

841 
842 
843 
844 
845 

70  72  81 
70  89  64 
71  06  49 
71  23  36 

71  40  25 

594  823  321 
596  947  688 
599  077  107 
601  211  584 
603  351  125 

29.OOOO 
29.0172 
29.0345 
29.0517 
29.0689 

9-4391 
9.4429 
9.4466 

9.4503 
9-4541 

846 
847 
848 
849 
850 

71  57  16 
71  74  09 
71  91  04 

72  08  01 
72  25  oo 

6°5  495  736 
607  645  423 
609  800  192 
6n  960  049 
614  125  ooo 

29.0861 
29.1033 
29.1204 
29.1376 
29.1548 

9-4578 
9.4615 
9.4652 
9.4690 

9-4727 

Number 

Square. 

Cube. 

Square  Root, 

Cube  Root. 

85I 
8S2 

853 
854 
855 

72  42  oi 
72  59  04 

72  76  09 

72  93  16 
73  I0  25 

616  295  051 
618  470  208 
620  650  477 
622  835  864 
625  026  375 

29.1719 
29.1890 
29.2062 
29.2233 
29.2404 

9.4764 
9.4801 
9.4838 

9-4875 
9.4912 

856 

857 
858 
8S9 
860 

73  27  36 
73  44  49 
73  61  64 
73  78  81 
73  96  oo 

627  222  Ol6 

629  422  793 
631  628  712 
633  839  779 
636  056  ooo 

29.2575 
29.2746 
29.2916 
29.3087 
29.^258 

9-4949 
9.4986 
9.5023 
9.5060 
9.5097 

86  1 
862 
863 
864 
865 

74  13  21 

74  3°  44 
74  47  69 
74  64  96 
74  82  25 

638  277  381 

640  503  928 

642  735  647 
644  972  544 
647  214  625 

29.3428 

29-3598 
29.3769 

29-3939 
29.4109 

9-5I34 
9.5171 

9.5207 
9.5244 
9.5281 

866 
867 
868 
869 
870 

74  99  56 
75  16  89 
75  34  24 
75  51  61 
75  69  oo 

649  461  896 
651  714  363 
653  972  032 
656  234  909 
658  503  ooo 

29.4279 

29.4449 
29.4618 
29.4788 
29.4958 

9-53'7 
9-5354 
9-539' 
9-5427 
9.5464 

871 
872 

873 
874 
875 

75  86  41 
76  03  84 
76  21  29 
76  38  76 
76  56  25 

660  776  311 
663  054  848 
665  338  617 
667  627  624 
669  921  875 

29.5127 
29.5296 
29.5466 

29-5635 
29.5804 

9-  55°  i 
9-5537 
9-5574 
9.5610 

9-  5647 

876 

877 
878 

879 
880 

76  73  76 
76  91  29 
77  08  84 
77  26  41 
77  44  oo 

672  221  376 
674  526  133 
676  836  152 

679  I51  439 

68  i  472  ooo 

29-5973 
29.6142 
29.6311 
29.6479 
29.6648 

9-5683 
9-57I9 
9-5756 
9-5792 
9.5828 

88  1 
882 
883 
884 
885 

77  61  61 
77  79  24 
77  96  89 
78  14  56 
78  32  25 

683  797  841 
686  128  968 
688  465  387 
690  807  104 
693  154  125 

29.6816 
29.6985 

29-7I53 
29.7321 
29.7489 

9-5865 
9.5901 

9-5937 
9-5973 
9.6010 

886 

887 
888 
889 
890 

78  49  96 
78  67  69 

78  85  44 

79  °3  2I 
79  21  oo 

695  5°6  456 
697  864  103 
700  227  072 
702  595  369 
704  969  ooo 

29.7658 
29.7825 
29.7993 
29.8161 
29.8329 

9.6046 
9.6082 
9.6118 
9.6154 
9.6190 

891 
892 

893 
894 
895 

79  38  81 
79  56  64 
79  74  49 
79  92  36 
80  10  25 

707  347  97i 
709  732  288 
712  121  957 
714  516  984 
716  917  375 

29.8496 
29.8664 
29.8831 
29.8998 
29.9166 

9.6226 
9.6262 
9.6298 

9-6334 

9.6370 

896 

897 
898 

899 
900 

80  28  16 
80  46  09 
80  64  04 
80  82  oi 
81  oo  oo 

7i9  323  *36 
721  734  273 
724  150  792 
726  572  699 
729  ooo  ooo 

29-9333 
29.9500 
29.9666 

29-9833 
3O.OOOO 

9.6406 
9.6442 
9.6477 
9-6513 
9.6549 

Number 

Square. 

Cube. 

Square  Root- 

Cube  Root. 

9OI 
902 

903 
904 

9°  5 

81  18  oi 
81  36  04 
81  54  09 
81  72  16 
81  90  25 

731  432  701 

733  870  808 
736  3H  327 
738  763  264 
741  217  625 

30.0167 
30-0333 
30.0500 
30.0666 
30.0832 

9-6585 
9.6620 
9.6656 
9.6692 
9.6727 

906 
907 
908 
909 
910 

82  08  36 
82  26  49 
82  44  64 
82  62  81 
82  81  oo 

743  677  4i6 
746  142  643 
748  613  312 
751  089  429 
753  57i  ooo 

30.0998 
30.1164 
30.1330 
30.1496 
30.1662 

9.6763 
9.6799 
9.6834 
9.6870 
9.6905 

9II 

9I2 

913 
914 

9*5 

82  99  21 

83  17  44 
83  35  69 
83  53  96 

83  72  25 

756  058  031 
758  550  825 
761  048  497 
763  551  944 
766  060  875 

30.1828 

3°-  '993 
30.2159 
30.2324 
30.2490 

9.6941 
9.6976 
9.7012 
9.7047 
9.7082 

916 
917 
918 
919 
920 

83  90  56 
84  08  89 
84  27  24 
84  45  61 
84  64  oo 

768  575  296 
771  095  213 
773  620  632 

776  151  559 
778  688  ooo 

30.2655 
30.2820 
30.2985 
30.3150 
30.3315 

9.7118 

9-7I53 
9.7188 
9.7224 
9.7259 

921 

922 

923 
924 

925 

84  82  41 
85  oo  84 

85  19  29 

85  37  76 
85  56  25 

781  229  961 
783  777  448 
786  330  467 
788  889  024 
791  453  125 

30.3480 

30.3645 
30.3809 

30.3974 
30.4138 

9.7294 
9.7329 
9-7364 

9.7400 

9-7435 

926 
927 
928 
929 
930 

85  74  76 
85  93  29 
86  ii  84 
86  30  41 
86  49  oo 

794  022  776 
796  597  983 
799  178  752 
801  765  089 
804  357  ooo 

30.4302 
30.4467 
30.4631 
30.4795 
30-4959 

9-7470 
9-7505 
9-7540 

9-7575 
9.7610 

93i 
932 
933 
934 
935 

86  67  61 
86  86  24 
87  04  89 
87  23  56 
87  42  25 

806  954  491 
809  557  568 
812  166  237 
814  780  504 
817  400  375 

30-5123 
30.5287 
30.5450 
30.5614 
30.5778 

9.7645 
9.7680 

9-77I5 

9-7750 
9-7785 

936 
937 
938 

939 
940 

87  60  96 
87  79  69 
87  98  44 
88  17  21 
88  36  oo 

820  025  856 
822  656  953 
825  293  672 
827  936  019 
830  584  ooo 

30-594I 
30.6105 
30.6268 
30.6431 
30.6594 

9-7819 
9.7854 
9.7889 
9.7924 
9-7959 

941 
942 
943 
944 
945 

88  54  81 
88  73  64 
88  92  49 
89  ii  36 

89  30  25 

833  237  621 

835  896  888 
838  561  807 
841  232  384 
843  908  625 

30.6757 
30.6920 
30.7083 
30.7246 
30.7409 

9-7993 
9.8028 
9.8063 
9.8097 
9.8132 

946 

947 
948 

949 

950 

89  49  16 
89  68  09 
89  87  04 
90  06  oi 
90  25  oo 

846  590  536 
849  278  123 
851  97i  392 

854  670  349 

857  375  ooo 

30.7571 
30.7734 
30.7896 
30.8058 
30.8221 

9.8167 
9.8201 
9.8236 

9.8270 
9.8.305 

Number 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

951 
952 

953 
954 
955 

90  44  01 
90  63  04 
90  82  09 
91  01  16 
91  20  25 

860  085  351 
862  80  I  408 
865  523  177 

868  250  664 
870  983  875 

30.8383 
30-8545 
30.8/07 
30.8869 
30.9031 

9.8339 
9-8374 
9.8408 
9.8443 
9.8477 

956 

957 
958 

959 
960 

91  39  36 
91  58  49 
91  77  64 
91  96  81 
92  16  oo 

873  722  816 
876  467  493 
879  217  912 
881  974  079 
884  736  ooo 

30.9192 
30.9354 
30.9516 
30.9677 
30.9839 

9.8511 
9.8546 
9.8580 
9.8614 
9.8648 

961 
962 

963 
964 
965 

92  3  21 
92  54  44 
92  73  69 
92  92  90 
93  12  25 

887  503  68  i 
890  277  128 

893  °56  347 
895  841  344 
898  632  125 

31.0000 
3I.Ol6l 
31.0322 
31.0483 
31.0644 

9.8683 
9.8717 
9.8751 
9.8785 
9.8819 

966 
967 
968 
969 
970 

93  3i  56 
93  5°  89 
93  70  24 
93  89  61 
94  09  oo 

901  428  696 
904  231  063 
907  039  232 
909  853  209 
912  673  ooo 

31.0805 
31.0966 
31.1127 
31.1288 
3LI448 

9.8854 
9.8888 
9.8922 
9.8956 
9.8990 

971 
972 
973 
974 
975 

94  28  41 
94  47  84 
94  67  29 
94  86  76 
95  06  25 

915  498  61  1 

918  330  048 
921  167  317 

924  oio  424 
926  859  375 

3I.l6o9 
31.1769 
31.1929 
31.2090 
31.2250 

9.9024 
9.9058 
9.9092 
9.9126 
9.9160 

976 

977 
978 

979 

980 

95  25  76 

95  45  29 
95  64  84 

95  84  41 
96  04  oo 

929  714  176 
932  574  833 
9jS  44i  352 
938  3i3  739 

941  192  ooo 

31.2410 
31.2570 
31.2730 
31.2890 

3^3°S° 

9.9194 
9.9227 
9,9261 
9.9295 
9.9329 

981 
982 

983 
984 
985 

96  23  61 
96  43  24 
96  62  89 
96  82  56 
97  02  25 

944  076  141 
946  966  i  68 
949  862  087 

952  763  904 
9<-5  671  625 

31.3209 

3I-3369 
3I-3528 
31.3688 
31-3847 

9'9363 
9.9396 

9-9430 
9.9464 

9-9497 

986 

987 
988 
989 
990 

97  21  96 
97  4i  69 
97  61  44 
97  81  21 
98  01  oo 

958  585  256 
961  504  803 
964  430  272 
967  361  669 
970  299  ooo 

31.4006 
31.4166 

3L4325 
31.4484 

31-4643 

9-9531 
9-956S 
9.9598 
9.9632 
9.9666 

991 
992 
993 
994 
995 

98  20  81 
98  40  64 
98  60  49 
98  80  36 
99  oo  25 

973  242  271 
976  191  488 
979  146  657 

982  107  784 
985  074  875 

31.4802 
31.4960 

3i-5II9 
31.5278 

3!-5436 

9-9699 
9-9733 
9.9766 
9.9800 
9.9833 

996 

997 
998 

999 

1000 

99  20  16 
99  40  09 
99  60  04 
99  80  01 

I  00  00  00 

988  047  936 
991  026  973 
994  on  992 
997  002  999 

I  OCO  OOO  OOO 

3L5595 
31*5753 

3I.59H 
31.6070 
31.6228 

9.9866 

9.9900 

9-9933 
9.9967 

IO.OOOO 

SHORT-TITLE     CATALOGUE 

OP  THE 

PUBLICATIONS 

OF 

JOHN   WILEY   &    SONS, 

NEW  YORK. 
LONDON:  CHAPMAN  &  HALL,  LIMITED. 


ARRANGED  UNDER  SUBJECTS. 


Descriptive  circulars  sent  on  application.  Books  marked  with  an  asterisk  (*)  are  sold 
at  net  prices  only,  a  double  asterisk  (**)  books  sold  under  the  rules  of  the  American 
Publishers'  Association  at  net  prices  subject  t«  an  extra  charge  for  postage.  •  All  booki 
are  bound  in  cloth  unless  otherwise  stated. 


AGRICULTURE. 

Annsby's  Manual  of  Cattle-feeding izrno,  Si  75 

Principles  of  Animal  Nutrition 8vo,    4  oo 

Budd  and  Hansen's  American  Horticultural  Manual: 


Part  I.  Propagation,  Culture,  and  Improvement  .................  i2mo, 

Part  II.  Systematic  Pomology  ................................  i2rro, 

Downing's  Fruits  and  Fruit-trees  of  America  ........................  8vo, 

Elliott's  Engineering  for  Land  Drainage  ...........................  i2mo, 

Practical  Farm  Drainage  ....................................  i2mo, 

Green's  Principles  of  American  Forestry  ...........................  i2mo, 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.)  ...........  i2mo, 

Kemp's  Landscape  Gardening  ....................................  i2mo, 

Maynard's  Landscape  Gardening  as  Applied  to  Home  Decoration  ......  i2mo, 


50 
50 

00 

50 

00 

50 

00 

50 


50 


Sanderson's  Insects  Injurious  to  Staple  Crops.  .  .....................  i2mo, 

Insects  Injurious  to  Garden  Crops.     (In  preparation.) 

Insects  Injuring  Fruits.     (In  preparation.) 

Stockbridge's  Rocks  and  Soils  ......................................  8vo,  2  50 

Woll's  Handbook  for  Farmers  and  Dairymen  .......................  i6mo,  i  50 

ARCHITECTURE. 

Baldwin's  Steam  Heating  for  Buildings  ............................  I2mo,  2  50 

Bashore's  Sanitation  of  a  Country  House  ..........................  1  21110.  i  oo 

Berg's  Buildings  and  Structures  of  American  Railroads  ................  4to,  5  oo 

Birkmire's  Planning  and  Construction  of  American  Theatres  ..........  .8vo,  3  oo 

Architectural  Iron  and  Steel.  ..................................  8vo,  3  50 

Compound  Riveted  Girders  as  Applied  in  Buildings  ................  8vo,  2  oo 

Planning  and  Construction  of  High  Office  Buildings  ...............  8vo  3  50 

Skeleton  Construction  in  Buildings  .............................  8vo,  3  oo 

Brigg's  Modern  American  School  Buildings  ..........................  8vo,  4  oo 

Carpenter's  Heating  and  Ventilating  of  Buildings  .....................  8vo.  4  oo 

Freitag's  Architectural  Engineering  .................................  8vo,  3  56 

Fireproofing  of  Steel  Buildings  .............................  ...  .8vo,  2  50 

French  and  Ives's  Stereotomy  .....................................  8vo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection  ......................  i6mo,  i  oo 

Theatre  Fires  and  Panics.  ,  ..................................  i2mo,  i  50 

Holly's  Carpenters'  and  Joiners'  Handbook  .........................  i8mo,  75 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods  ..................  8vo,  2  oo 


Kidder's  Architects' and  Builders' Pocket-book.  Rewritten  Edition.  i6mo,mor.,  5  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,    5  oo 

Non-metallic  Minerals:   Their  Occurrence  and  Uses 8vo,    4  oo 

Monckton's  Stair- building 4to,    4  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,    5  oo 

Peabody's  Naval  Architecture Svo,    7  50 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor  ,    4  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,    3  oo 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,    i  50 

Snow's  Principal  Species  of  Wood 8vo,    3  50 

Sondericker's  Graphic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,    2     3 

Towne's  Locks  and  Builders'  Hardware i8mo,  morocco,    3  oo 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,    6  oo 

Sheep,    6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,    5  oo 

Sheep,    5  50 

Law  of  Contracts 8vo,    3  oo 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,    4  oo 

Woodbury's  F're  Protection  of  Mills 8vo,    2  50 

Worcester  and  Atkinson's  Small  Hospitals,  Establishment  and  Maintenance, 
Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

•  i2mo,     i  25 
The  World's  Columbian  Exposition  of  1893 Large  4to,    i  oo 

ARMY  AND  NAVY. 

Bernadou's  Smokeless  Powder,  Nitro-cellulose,  and  the  Theory  of  the  Cellulose 

Molecule i2mo,    2  50 

*  Bruff  *s  Text-book  Ordnance  and  Gunnery 8vo,    6  oo 

Chase's  Screw  Propellers  and  Marine  Propulsion 8vo,    3  oo 

Cloke's  Gunner's   Examiner 8vo,     i   50 

Crarg's  Azimuth 4to,  3  50 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  oo 

Cronkhite's  Gunnery  for  Non-commissioned  Officers 241110,  morocco,  2  oo 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,    7  oo 

Sheep,  7  So 

De  Brack's  Cavalry  Outposts  Duties.     (Carr.) 24mo,  morocco,  2  oo 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco,  i  25 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco,  15  oo 

Durand's  Resistance  and  Propulsion  of  Ships 8vo,  5  oo 

*  Dyer's  Handbook  of  Light  Artillery I2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

*  Fiebeger's  Text-book  on  Field  Fortification Small  8vo,  2  oo 

Hamilton's  The  Gunner's  Catechism i8mo,  i  oo 

*  Hoff's  Elementary  Naval  Tactics 8vo,  i  50 

Ingalls's  Handbook  of  Problems  in  Direct  Fire 8vo,  4  oo 

*  Ballistic  Tables 8vo,    i  50 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols.  I.  and  II.  .8vo,  each,    6  oo 

*  Mahan's  Permanent  Fortifications.    (Mercur.) .  .  . : 8vo,  half  morocco,    7  50 

Manual  for  Courts-martial i6mo,  morocco,     i  50 

*  Mercur's  Attack  of  Fortified  Places i2mo,    2  oo 

?•&£  £     Elements  of  the  Art  of  War 8vo,    4  oo 

Metcalf's  Cost  of  Manufactures — And  the  Administration  of  Workshops.  .8vo,  5  oo 

*  Ordnance  and  Gunnery.     2  vols i2mo,  5  oo 

Murray's  Infantry  Drill  Regulations i8mo,  paper,  10 

Hixon's  Adjutants'  Manual 24010,  i  oo 

Peabody's  Naval  Architecture 8vo,  7  50 

2 


*  .rhelps's  Practical  Marine  Surveying 8vo,  2  50 

Powell's  Army  Officer's  Examiner 1210.0,  4  oo 

Sharpe's  Art  of  Subsisting  Armies  in  War ". i8mo.  morocco,  i  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

*  Wheeler's  Siege  Operations  and  Military  Mining 8vo,  2  oo 

Winthrop's  Abridgment  of  Military  Law , i2mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 

Young's  Simple  Elements  of  Navigation i6mo,  morocco,  i  oo 

Second  Edition,  Enlarged  and  Revised i6mo,  morocco,  2  oo 

ASSAYING. 
Fletcher's  Practical  Instructions  ir*  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  .  .  .8vo,  3  oo 

Miller's  Manual  of  Assaying i2mo,  i  oo 

O'Driscoli's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process i2mo,  i  50 

ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Craig's  Azimuth 4to,  3  50 

Doolittle's  Treatise  on  Practical  Astronomy 8vo,  4  oo 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

*  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  oo 

*  White's  Elements  of  Theoretical  and  Descriptive  Astronomy i2mo,  2  oo 

BOTANY. 

Davenport's  Statistical  Methods,  with  Special  Reference  to  Biological  Variation. 

i6mo,  morocco,     i  25 

Thome'  and  Bennett's  Structural  and  Physiological  Botany i6mo,    2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.) 8vo,    2  oo 

CHEMISTRY. 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables I2mo,  i  25 

Allen's  Tables  for  Iron  Analysis 8vo,  3  oo 

Arnold's  Compendium  of  Chemistry.     (Mandel.) Small  8vo,  3  50 

Austen's  Notes  for  Chemical  Students i2mo,  i  50 

Bernadou's  Smokeless  Powder. — Nitro-cellulose,  and  Theory  of  the  Cellulose 

Molecule i2tno,  2  50 

Bolton's  Quantitative  Analysis 8vo,  i  50 

*  Browning's  Introduction  to  the  Rarer  Elements. 8vo,  i  50 

Brush  and  Penfield's  Manual  of  Determinative  Mineralogy 8vo,  4  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Boltwood.).  .8vo,  3  oo 

Cohn's  Indicators  and  Test-papers I2mo,  2  oo 

Tests  and  Reagents 8vo,  3  oo 

Crafts's  Short  Course  in  Qualitative  Chemical  Analysis.   (Echaeffer.). .  .i2mo,  i  50 
Dolezalek's  Theory  of  the  Lead  Accumulator   (Storage  Battery).        (Von 

Ende.) I2mo,  2  50 

Drechsel's  Chemical  Reactions.     (Merrill.) i2mo,  i  25 

Duhem's  Thermodynamics  and  Chemistry.     (Eurgess.) 8vo,  4  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) i2mo,  i  25 


Fletcher's  Practical  Instructions  im  Quantitative  Assaying  with  the  Blc*  pipe. 

i2mo,  morocco,  i  50 

Fowler's  Sewage  Works  Analyses i2mo,  2  oo 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,  5  oo 

Manual  of  Qualitative  Chemical  Analysis.  Part  I.  Descriptive.  (Wells.)  8vo,  3  oo 
System   of    Instruction   in    Quantitative    Chemical   Analysis.      (Cchn.) 

2  vols 8vo,  12  50 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

*  Getman's  Exercises  in  Physical  Chemistry i2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i   25 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) i2mo,  2  oo 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (Mandel.) 8vo,  4  oo 

Helm's  Principles  of  Mathematical  Chemistry.     (Morgan.) i2mo,  i  50 

Bering's  Ready  Reference  Tables  (Conversion  Factors) iCn  o  rrcrccco,  2  50 

Hind's  Inorganic  Chemistry 8vo,  3  oo 

*  Laboratory  Manual  for  Students i2mo,  i  oo 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) 8vo,  2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott.) 8vo,  2  50 

*  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) i2mo,  i  co 

Hopkins*  s  Oil-chemists'  Handbook 8vo,  3  oo 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo,  i  25 

Keep's  Cast  Iron 8vo,  2  50 

Ladd's  Manual  of  Quantitative  Chemical  Analysis i2mo,  i  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

*  Langworthy  and   Austen.        The   Occurrence  of  Aluminium  in  Vege  able 

Products,  Animal  Products,  and  Natural  Waters 8vo,  2  oo 

Lassar-Cohn's  Practical  Urinary  Analysis.  (Lorenz.) i2mo,  i  oo 

Application  of  Some  General  Reactions  to  Investigations  in  Organic 

Chemistry.  (Tingle.) i2rr.o,  i  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz. ).i2iro,  i  co 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments 8vo,  3  co 

Lunge's  Techno-chemical  Analysis.  (Cohn.) i2mo,  i  co 

Mandel's  Handbook  for  Bio-chemical  Laboratory i2mo,  i  50 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe .  .  i2rro,  Co 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo,  ,  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2rro,  i  25 

Matthew's  The  Textile  Fibres 8vo,  3  so 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  .  i2mo,  i  oo 

Miller's  Manual  of  Assaying i2mo,  i  oo 

Mixter's  Elementary  Text-book  of  Chemistry i2mo,  i  50 

Morgan's  Outline  of  Theory  of  Solution  and  its  Results i2mo,  i  oo 

Elements  of  Physical  Chemistry i2mo,  2  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Ccrr  pounds. 

Vol.  I Large  8vo,  5  oo 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,  2  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ostwald's  Conversations  on  Chemistry.     Part  One      (Ramsey.) i2mo,  150 

Ostwald's  Conversations  on  Chemistry.     Part  Two.     (Turnbull  ).     (In  Press.) 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mireral  Tests. 

8vo,  paper,  50 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) 8vo,  5  oo 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) i2mo,  i  50 

Pooie's  Calorific  Power  of  Fuels 8vo,  3  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  i  23 

4 


*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Standpoint  8vo,  2  oo 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  oo 

Cost  of  Food,  a  Study  in  Dietaries i2mo,  i  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic  Chemistry.     (Part  I. 

Non-metallic  Elements.) 8vo,  morocco,  75 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Disinfection  and  the  Preservation  of  Food 8vo,  4  oo 

Rigg's  Elementary  Manual  for  the  Chemical  Laboratory 8vo,  i  25 

Rostoski's  Serum  Diagnosis.  (Bolduan.) i2mo,  i  oo 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Salkowski's  Physiological  and  Pathological  Chemistry.  (Orndorff.) 8vo,  2  50 

Schimpf's  Text-book  of  Volumetric  Analysis 1 2mo,  2  50 

Essentials  of  Volumetric  Analysis i2mo,  i  25 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Sugar  Manufacturers  and  their  Chemists.  .  i6mo,  morocco,  2  oo 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

*  Descriptive  General  Chemistry 8vo,  3  oo 

Treadwell's  Qualitative  Analysis.     (Hall.) 8vo,  3  oo 

Quantitative  Analysis.     (Hall.) 8vo,  4  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2mo,  i  50 

*  Walke's  Lectures  on  Explosives 8"o,  4  oo 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8-0,  2  oo 

Wassermann's  Immune  Sera :  Haemolysins,  Cytotoxins,  and  Precipitins.    (Bol- 
duan.)   i2mo,  i  oo 

Well's  Laboratory  Guide  in  Qualitative  Chemical  Analysis 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students I2mo,  i  50 

Text-book  of  Chemical  Arithmetic i2mo,  i  25 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Cyanide  Processes : I2mo,  i  50 

Chlorination  Process 12010,  i  50 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry I2mo,  2  oo 

CIVIL  ENGINEERING. 

BRIDGES    AND    ROOFS.       HYDRAULICS.       MATERIALS    OF    ENGINEERING. 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments i2mo,  3  oo 

Bixby's  Graphical  Computing  Table Paper  19^X24$  inches.  25 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal.     (Postage, 

27  cents  additional.) 8vo,  3  50 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Davis's  Elevation  and  Stadia  Tables 8vo,  i  oo 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  50 

Practical  Farm  Drainage i2mo,  i  oo 

*Fiebeger's  Treatise  on  Civil  Engineering 8vo,  5  oo 

Folwell's  Sewerage.     (Designing  and  Maintenance.) 8vo,  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Goodhue's  Municipal  Improvements I2mo,  i  75 

Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo,  3  50 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

5 


Howe's  Retaining  Walls  for  Earth i2mo,  i  25 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  oo 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods .8vo,  2  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.)  -  i2mo,  2  oo 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo,  5  oo 

*  Descriptive  Geometry 8vo,  i  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

Elements  of  Sanitary  Engineering 8vo,  2  oo 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco,  2  oo 

Nugent's  Plane  Surveying 8vo,  3  50 

Ogden's  Sewer  Design i2mo,  2  oo 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillarx) 8vo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Learns,  and  Arches. 

8vo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  oo 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo ,  5  oo 

Sheep,  s  50 

Law  of  Contracts 8vo,  3  oo 

Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,  2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,  i  25 

*  Wheeler  s  Elementary  Course  of  Civil  Engineering 8vo,  4  oo 

Wilson's  Topographic  Surveying -. 8vo,  3  50 

BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges .  .  8ro,  2  oo 

*  Thames  River  Bridge 4to,  paper,  5  oo 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges 8vo,  3  50 

Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations.  .  .  .8vo,  3  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  oo 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Greene's  Roof  Trusses 8vo,  i  25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Howe's  Treatise  on  Arches 8vo,  4  oo 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel 8vo,  2  oo 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,  10  oo 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I.     Stresses  in  Simple  Trusses , 8vo,  2  50 

Part  H.     Graphic  Statics 8vo,  2  50 

Part  III.     Bridge  Design 8vo,  2  50 

Part  IV.     Higher  Structures , 8vo,  2  50 

Morison's  Memphis  Bridge 4to,  10  oo 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  morocco,  3  oo 

Specifications  for  Steel  Bridges i2mo.  i  25 

Wood's  Treatise  on  the  Theory  of  the  Construction  of  Bridges  and  Roofs.  .8vo,  2  CO 
Wright's  Designing  of  Draw-spans : 

Part  I.     Plate-girder  Draws 8vo,  2  50 

Part  II.     Riveted-truss  and  Pin-connected  Long-span  Draws 8vo,  2  50 

Two  parts  in  one  volume 8vo,  3  50 


HYDRAULICS. 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) 8vo,  2  oo 

Bovey's  Treatise  on  Hydraulics 8vo,  5  op 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i  50 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Folwell's  Water-supply  Engineering 8vo,  4  oo 

Frizell's  Water-power ' 8vo,  5  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) 8vo,  4  oo 

Hazen's  Filtration  of  Public  Water-supply 8vo,  3  oo 

Hazlehurst's  Towers  and  Tanks  for  Water-works 8vo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo,  2  oo 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

8vo,  4  oo 

Merriman's  Treatise  on  Hydraulics 8vo,  5  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Schuyler's   Reservoirs   for   Irrigation,   Water-power,   and   Domestic   Water- 
supply Large  8vo,  5  oo 

**  Thomas  and  Watt's  Improvement  of  Rivers.     (Post.,  44C.  additional.). 4*0,  6  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Wegmann's  Design  and  Construction  of  Dams 4*0,  5  oo 

Water-supply  of  the  City  of  New  York  from  1638  to  1895 4to,  10  oo 

Williams  and  Hazen's  Hydraulic  Tables 8vo,  i  50 

Wilson's  Irrigation  Engineering Small  8vo,  4  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics 8vo,  3  oo 

MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  «n  Masonry  Construction 8vo,  5  oo 

Roads  and  Pavements 8vo,  5  oo 

Black's  United  States  Public  Works Oblong  410,  5  oo 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  50 

Byrne's  Highway  Construction 8vo,  5  oo 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  4to,  7  50 

*Eckel's  Cements,  Limes,  and  Plasters 8vo,  6  oo 

Johnson's  Materials  of  Construction.  .  ." Large  8vo,  6  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Marten's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo,  7  50 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Merriman's  Mechanics  of  Materials.                                  8vo,  5  oo 

Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users I2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Richardson's  Modern  Asphalt  Pavements 8vo,  3  oo 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor.,  4  oo 

Rockwell's  Roads  and  Pavements  in  France i2mo,  i  23 

7 


Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vof  3  oa 

Smith's  Materials  of  Machines I2mo,  i  oo 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Text-book  on  Roads  and  Pavements i2mo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Materials  of  Engineering.     3  Parts 8vo,  8  oo 

Part  I.     Non-metallic  Materials  of  Engineering  and  Metallurgy 8vo,  2  oo 

Part  II.     Iron  and  Steel ' 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents. .._ 8vo,  2  50 

Thurston's  Text-book  of  the  Materials  of  Construction 8vo,  5  oo 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  oo 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.)-  •  i6mo,  mor.,  3  oo 

Specifications  for  Stc  i  Bridges i2mo,  i  25 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood'-  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,  4  oo 

RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Brook's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  morocco,  5  oo 

Dredge's  History  of  the  Pennsylvania  Railroad:   (1879) Paper,  5  oo 

*  Drinker's  Tunnelling,  Explosive  Compounds,  and  Rock  Drills'. 4to,  half  mor.,  25  oo 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide.  .  .  i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  o*f  Excavations  and  Em- 
bankments.   8vo,  i  oo 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  oo 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  oo 

Searles's  Field  Engineering i6mo,  morocco,  3  oo 

Railroad  Spiral i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  i  50 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  oo 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

^       i2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  oo 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  5  oo 

DRAWING. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "                    "                    "        Abridged  Ed 8vo,  150 

Coolidge's  Manual  of  Drawing 8vo,  paper  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo.  2  50 

8 


Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  oo 

Jamison's  Elements  of  Mechanical  Drawing 8vo,  2  50 

Advanced  Mechanical  Drawing .- 8vo,  2  oo 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

MacCord's  Elements  of  Descriptive  Geometry 8vo,  3  oo 

Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting 8vo,  i  50 

Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Moyer's  Descriptive  Geometry 8vo,  2  oo 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Robinson's  Principles  of  Mechanism 8vo»  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo, 


Drafting  Instruments  and  Operations i2mo. 

Manual  of  Elementary  Projection  Drawing i2mo, 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo, 

Plane  Problems  in  Elementary  Geometry i2mo, 


oo 
25 
5» 

00 

25 

Primary  Geometry i2mo,  75 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  50 

General  Problems  of  Shades  and  Shadows 8vo,  3  oo 

Elements  of  Machine  Construction  and  Drawing.  .  .. 8vo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry.  ....  .'.8vo,  2  50 

Weisbach's  Kinematics  and  Power  of  Transmission.    (Hermann  and  Klein)8vo,  5  oo 

Whelpley's  Practical  Instruction  in  the  Ait  of  Letter  Engraving i2mo,  2  oo 

Wilson's  (H.  M.)  Topographic  Surveying .8vo,  3  50 

Wilson's  (V.  T.)  Free-hand  Perspective 8vo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering 8vo,  i  oo 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  oo 


ELECTRICITY  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo,  3  oo 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  .  .  .  i2mo,  i  oo 

Benjamin's  History  of  Electricity, 8vo,  3  oo 

Voltaic  Cell 8vo,  3  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).8vo,  3  oo 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  oo 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco,  5  oo 
Dolezalek's    Theory   of   the    Lead   Accumulator    (Storage    Battery).      (Von 

Ende.) I2mo,  2  50 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Gilbert's  De  Magnete.     (Mottelay.) 8vo,  2  50 

Hanchett's  Alternating  Currents  Explained I2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2.  50 

Holman's  Precision  of  Measurements : 8vo,  2  oo 

Telescopic   Mirror-scale  Method,  Adjustments,  and  Tests.  .  .  .Large  8vo,  75 

Kinzbrunner's  Testing  of  Continuous-Current  Machines 8vo.  2  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

Le  Chatelien's  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo,  3  oo 

Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz.)  12 mo,  i  oo 


*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols.  I.  and  II.  8vo,  each,    6  oo 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,    4  oo 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo, 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  .  .8vo, 


Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     VoL  1 8vo, 

Thurston's  Stationary  Steam-engines 8vo, 

*  Tillman's  Elementary  Lessons  in  Heat 8vo, 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo, 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

LAW. 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,  7  oo 

*  Sheep,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

Winthrop's  Abridgment  of  Military  Law 121110,  2  50 

MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  5* 

Bolland's  Iron  Founder i2mo,  2  50 

"The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding '. i2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Fitzgerald's  Boston  Machinist i2mo,  i  oo 

Ford's  Boiler  Making  for  Boiler  Makers '. i8mo,  i  oo 

Hopkin's  Oil-chemists'  Handbook 8vo,  3  oo 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control. Large  8vo,  7  50 

Matthews's  The  Textile  Fibres 8vo,  3  50 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Mstcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops  8vo,  5  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Spalding's  Hydraulic  Cement i2tno,  2  oo 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses.    ...  i6mo,  morocco,  3  oo 

Handbook  for  Sugar  Manufacturers  and  their  Chemists.  .  i6mo,  morocco,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  8vo,  5  oo 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Ware's  Manufacture  of  Sugar.     (In  press.) 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

10 


Wolff's  Windmill  as  a  Prime  Mover  . ' 8vo,    3  oo 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,    4  oft 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo, 

*  Bass's  Elements  of  Differential  Calculus i2mo, 

Briggs's  Elements  of  Plane  Analytic  Geometry I2mo," 

Compton's  Manual  of  Logarithmic  Computations i2mo, 

Davis's  Introduction  to  the  Logic  of  Algebra : 8vo, 

*  Dickson's  College  Algebra Large  i2mo, 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  12 mo, 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo, 

Halsted's  Elements  of  Geometry 8vo, 

Elementary  Synthetic  Geometry . .  .8vo, 


5® 

00 

oo 
50 

50 
50 
25 
50 
75 
50 


Rational  Geometry i2mo,  75 

*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:   Vest-pocket  size. paper,  15 

100  copies  for  5  oo 

Mounted  on  heavy  cardboard,  8X  10  inches,  25 

10  copies  for  2  oo 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.  .Small  8vo,  3  oo 

Johnson's  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus .  Small  8vo,  i  50 

Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo,  i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  a  ad  Partial  Differential  Equations. 

Small  8vo,  3  50 

Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares  i2mo,  i  50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics 1200,  3  oo 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.) .  i2mo,  2  oo 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,  3  oo 

Trigonometry  and  Tables  published  separately Each,  2  oo 

*  Ludlow's-Logarithmic  and  Trigonometric  Tables 8vo,  i  oo 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman  and  Woodward's  Higher  Mathematics 8vo,  5  oo 

Merriman's  Method  of  Least  Squares. . .  .*. 8vo,  2  oo 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  8vo,  3  oo 

Differential  and  Integral  Calculus.     2  vols.  in  one Small  8vo,  2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,  2  oo 

Trigonometry:  Analytical,  Plane,  and  Spherical i2mo,  i  oo 


MECHANICAL  ENGINEERING. 

MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  50 

Barr's  Kinematics  of  Machinery. 8vo,  50 

*  Bartlett's  Mechanical  Drawing • 8vo,  oo 

Abridged  Ed 8vo,  50 

Benjamin's  Wrinkles  and  Recipes i2mo,  oo 

Carpenter's  Experimental  Engineering 8vo,  6  oo 

Heating  and  Ventilating  Buildings 8vo,  4  oo 

Cary's  Smoke  Suppression  in  Plants  using  Bituminous  Coal.     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,  4  oo 

Coolidge's  Manual  of  Drawing 8vo,  paper,  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

11 


Cromwell's  Treatise  on  Toothed  Gearing izmo,  I  50 

Treatise  on  Belts  and  Pulleys I2mo,  i  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Flather's  Dynamometers  and  the  Measurement  of  Power :  i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers 12010,  i  25 

Hall's  Car  Lubrication i2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Button's  The  Gas  Engine 8vo,  5  oo 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco,  5  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.)  .  .  8vo,  4  oo 

MacCord's  Kinematics;   or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

Mahan's  Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Poole  s  Calorific  Power  of  Fuels 8vo,  3  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richard's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Thurston's   Treatise   on   Friction  and   Lost   Work   in   Machinery  and   Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  12 mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — 

Klein.) 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  oo 

Wolff's  Windmill  as  a  Prime  Mover •. 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

MATERIALS   OF    ENGINEERING. 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.    6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Johnson's  Materials  of  Construction 8vo,  6  oo 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Merriman's  Mechanics  of  Materials.                                8vo,  5  oo 

Strength  of  Materials I2mo,  i  oo 

Metcalf 's  SteeL     A  manual  for  Steel-users I2mo.  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines I2mo,  i  oo 

rhurston's  Materials  of  Engineering 3  vols.,  8vo,  8  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  HI.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Text-book  of  the  Materials  of  Construction SVQ,  5  oo 

12 


Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preseivation  of  Timber 8vo,    2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,    3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Sted. 8vo,    4  oo 


STEAM-ENGINES  AND  BOILERS. 


Berry's  Temperature-entropy  Diagram 12010,  23 

Carnot's  Reflections  on  the  Motive  Power  «f  Heat.     (Thurston.) i2mo,  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  .  .i6mo,  mor.,  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  oo 

Goss's  Locomotive  Sparks •. 8vo,  2  oo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  oo 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,  5  oo 

Heat  and  Heat-engines 8vo,  5  oo 

Kent's  Steam  boiler  Economy 8vo,  4  oo 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  i  50 

MacCord's  Slide-valves , ' .8vo,  2  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Peabody's  Manual  of  the  Steam-engine  Indicator I2mo.  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors 8vo,  i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,  5  oo 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers : 8vo,  4  oo 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  i  25 

Reagan's  Locomotives:  Simple   Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  oo 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo,  2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice I2mo,  2  50 

Snow's  Steam-boiler  Practice 8vo,  3  oo 

Spangler's  Valve-gears 8vo,  2  50 

Notes  on  Thermodynamics izmo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Handy  Tables 8vo.  i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  oo 

Part  I.     History,  Structure,  and  Theory 8vo,  6  oo 

Part  H.     Design,  Construction,  and  Operation 8vo,  6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  oo 

Stationary  Steam-engines 8vo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice i2mo,  i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Cperation 8vo,  5  oo 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,  5  oo 

Whitham's  Steam-engine  Design 8vo,  5  oo 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo,  2  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .8vo,  4  oo 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Chase's  The  Art  of  Pattern-making <.....  I2mo,  2  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

13 


Church's  Notes  and  Examples  in  Mechanics 8vo,  oo 

Compton's  First  Lessons  in  Metal-working izmo,  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  50 

Treatise  on  Belts  and  Pulleys i2mo,  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools .  .  i2mo,  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  oo 

Dredge's  Record  of  the  Transportation  Exhibits  Building  of  the   World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  oo 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    II.     Statics 8vo,  4  oo 

Vol.  III.     Kinetics 8vo,  3  50 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

VoL  II Small  4to,  10  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Fitzgerald's  Boston  Machinist i6mo,  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving I2mo,  2  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Hall's  Car  Lubrication T i2mo,  i  oo 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle.  Srn.8vo,2  oo 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vc,  3  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Lanza's  Applied  Mechanics 8vo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*Lorenz's  Modern  Refrigerating  Machinery.      (Pope,  Haven,  and  Dean.). 8vo,  4  oo 

MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Elements  of  Mechanics i2mo,  i  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Reagan's  -Locomotives:  Simple,  Compound,  and  Electric i2mo,  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     VoL  1 8vo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Sinclair's  Locomotive-engine  Running  and  Management 12 mo,  2  oo 

Smith's  (0.)  Press-working  of  Metals 8vo,  3  oo 

Smith's  (A.  W.)  Materials  of  Machines I2mo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Treatise  on  Friction  and  Lost  V/ork  in    Machinery  and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

i2mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.   ( Herrmann — Klein. ) .  8vo ,  5  oo 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein. ).8vo,  5  oo 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Principles  of  Elementary  Mechanics I2mo,  i  25 

Turbines 8vo ,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

14 


METALLURGY. 

Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    L     Silver 8vo,  7  SO 

Vol.  II.     Gold  and  Mercury. 8vo,  7  So 

**  Iles's  Lead-smelting.     (Postage  9  cents  additional.) I2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe • 8vo,  i  go 

Le  Chatelier's  High-temperature  Measuremepts.  (Boudouard— Burgess. )i2mo,  3  oo 

Metcalf  s  SteeL     A  Manual  for  Steel-users-     i2mo,  2  oo 

Smith's  Materials  of  Machines i2mo,  i  oo 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo.  8  oo 

Part    II.     Iron  and  SteeL 8vo,  3  5<> 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Hike's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  oo 

Map  of  Southwest  Virignia Pocket-book  form.  2  oo 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo.  4  oo 

Chester's  Catalogue  of  Minerals 8vo,  paper,  i  oo 

Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals 8vo,  3  50 

Dana's  System  of  Mineralogy Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  8vo,  i  oo 

Text-book  of  Mineralogy 8vo,  4  oo 

Minerals  and  How  to  Study  Them I2mo,  i  50 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  i  oo 

Manual  of  Mineralogy  and  Petrography i2mo,  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  oo 

Eakle's  Mineral  Tables 8vo,  i  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.). Small  8vo,  2  oo 

Merrill's  Non-metallic  Minerals:   Their  Occurrence  and  Uses 8vo,  4  oo 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo  paper,  o  50 
Rosenbusch's   Microscopical  Physiography   of   the   Rock-making  Minerals. 

(Iddings.) 8vo.  5  oo 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks .8vo.  2  oo 

Williams's  Manual  of  Lithology 8vo,  3  oo 

MINING. 

Beard's  Ventilation  of  Mines I2mo.  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo.  3  oo 

Map  of  Southwest  Virginia Pocket  book  form.  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects  . i2mo.  i  oo 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.   4to,hf.  mor..  25  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Fowler's  Sewage  Works  Analyses. I2tno,  2  oo 

Goodyear 's  Coal-mines  of  the  Western  Coast  of  the  United  States i2mo.  2  50 

Ihlseng's  Manual  of  Mining . 8vo.  5  oo 

**  Iles's  Lead-smelting.     (Postage  oc.  additional.) I2mo.  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe .8ya,  i  50 

O'DriscolTs  Notes  on  the  Treatment  of  Gold  Ores 8vo.  2  oo 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Wilson's  Cyanide  Processes _ I2mo,  i  50 

Chlorination  Process .  i2mo,  i  50 

15 


Wilson's  Hydraulic  and  Placer  Mining I2mo,  2  oo 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation tamo',  i  25 

SANITARY  SCIENCE. 

Bashore's  Sanitation  of  a  Country  House I2mo,  i  oo 

FolwelTs  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo,  3  oc 

Water-supply  Engineering gvo,  4  oo 

Fuertes's  Water  and  Public  Health. i2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  8vo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control gvo,  7  50 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Merriman's  Elements  of  Sanitary  Engineering 8vo,  2  oo 

Ogden's  Sewer  Design i2mo,  2  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  i  25 

*  Price's  Handbook  on  Sanitation i2mo,  i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,  i  oo 

Cost  of  Living  as  Modified  by  Sanitaiy  Science iamo,  i  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Stand- 
point  8vo,  2  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 

MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.  (Rosanoff  and  Collins.).  ..  .Large  i2mo,  2  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo.  4  oo 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  ot  Food.  Mounted  chart,  i  25 

Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  oo 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894.  .Small  8vo,  3  oo 

Rostoski's  Serum  Diagnosis.  (Bolduan.) i2mo,  i  oo 

Rotherham's  Emphasized  New  Testament Large  8vo,  2  oo 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Totten's  Important  Question  in  Metrology 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

Von  Behring's  Suppression  of  Tuberculosis.  (Bolduan.) 121110,  i  oo 

Winslow's  Elements  of  Applied  Microscopy i2mo,  i  50 

Worcester  and  Atkinson.  Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture :  Plans  for  Small  Hospital.  i2mo,  i  25 

HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  3  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  tr   the  Old  Testament  Scriptures. 

(Tregelles.) Small" 4to,  half  morocco,  5  oo 

Uttews's  Hebrew  Bible 8vo,  2  25 

16 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

ASTRONOMY,- MATHEMATICS 


This 


tamped  below,  or 
on  the  date  to  wfiicK  "renewed. 
Renewed  books  are  subject  to  immediate  recall. 


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